Heuristic inspection method to find causes of system abnormality based on dynamic uncertain causality graph

ABSTRACT

Provided is a method for ordering X-type variables having states to be measured in a dynamic uncertain causality graph (DUCG). The method comprises: step 1: determining, on the basis of a DUCG simplified using E(y) as a condition, a measurable X-type variable having a state to be measured, an index set thereof being S x(y), where y is a time series; step 2: if there is only one element in S x(y), ending ordering; step 3: calculating ranking importance I i(y) for X i(i∈S x(y)); step 4: ranking X i(i∈S x(y)) according to the ranking importance I i(y), and performing state measurement on an X-type variable of i∈S x(y) by referring to the ranking; and step 5: adding 1 to y, and repeating steps 1-5 until there is no X-type variable to be measured. The technical solution of the invention can quickly diagnose a cause of an object system abnormality at minimum cost, and effectively returns the object system to normal.

FIELD OF THE INVENTION

This invention is about an AI technology to process uncertain causalitytype information represented by Dynamic Uncertain Causality Graph(DUCG). Based on the technical scheme proposed in this invention,through computation with computer, one can rank the variables to bedetected optimally, detect the ranked variables sequentially or group bygroup to find out their states, to diagnose causes of system abnormalityas earlier and less cost as possible, and then enable to take effectiveactions to make system normal.

BACKGROUND OF THE INVENTION

There exist many cause events that may lead to abnormalities ofindustrial systems, social systems and biological systems, such as shortcircuit of coils, pump fails to run, failure of components, malfunctionof sub-systems, blocking of pathway, entry of foreign matter, pollution,infection, damage, decrepitude of tissue or organ, etc. When systemabnormality appears, people need to know the real cause event as soon aspossible. Denote B_(n) or BX_(n) as these cause event variables indexedby n and B_(nk) or BX_(nk) as the event that B_(n) or BX_(n) is in statek. The difference between B_(n) and BX_(n) is that B_(n) representsroot-cause variable without input while BX has inputs reflecting theinfluence of other factors. Usually, k=0 means that B_(n) or BX_(n) isin normal state; k=1, 2, 3 . . . means that B_(n) or BX_(n) is inabnormal state. Most of the states of B_(n) and BX cannot be or are hardto be detected directly. Furthermore, there are a large number offactors that have causal relationships to B_(n) or BX_(n), such astemperature, pressure, flow, velocity, frequency, various chemical orphysical test results, investigation results, imaging examinationresults, acoustic examination results, and so on. Some factors mayincrease or decrease the occurrence probability of B_(nk) (k≠0), e.g.region, time, environment, season, religion, age, sex, skin color,career, blood relation, hobbies, personality, living conditions, workingconditions, etc. The affected B_(nk) is BX_(nk). All these factors canbe represented by event variable X_(i), while X_(ij) represents state jof X_(i). X_(i) and X_(ij) also represent cause or consequence of othervariables/events. Usually, j=0 means that X is in normal state; j=1, 2,3 . . . means that X is in abnormal state. By detecting the states ofX_(i), people can diagnose the root cause (B_(nk) or BX_(nk) (k≠0)) ofsystem abnormality, so that be able to take effective actions in time toget the system back to normal or reduce damage. Dynamic UncertainCausality Graph (DUCG) is an AI scheme to represent the uncertain causalrelationships among event variables graphically, and perform diagnosesbased on the constructed graph and observed evidence E composed of knownstates of X-type variables X_(ij). For example,E=X_(1,2)X_(2,3)X_(3,1)X_(4,0)X_(5,0), where comma separates thevariable index (the first) and state index (the second). In general, themore the state known X-type variables in E, the more accurate thediagnosis can be. Yet some state known X-type variables contributelittle or none to the diagnosis, some contribute a lot. In practice,there exist cost to detect the states of X-type variables. Therefore,one may have to choose some to detect, or detect some earlier and somelater. The problem to be solved in this invention is as follows:

Based on

-   -   (1) the constructed DUCG,    -   (2) evidence E(y) observed by the time indexed with y (y=0, 1,        2, . . . ),    -   (3) possible cause events included in hypothesis set S_(H)(y)        diagnosed conditional on E(y),        which state-unknown and detectable X-type variable should be        detected to update evidence E(y) as E(y+1)=E⁺(y)E(y), so that        more accurate diagnoses S_(H)(y+1) conditional on E(y+1) can be        made as soon and with less cost as possible. Where, the new        detected states of X-type variables are the new evidence denoted        as E⁺(y), the new hypothesis set denoted as S_(H)(y+1) is        diagnosed conditional on E(y+1), and y=0 means the time no        evidence received, that is, E(0) is a complete set denoted as        E(0)=1.

The possible cause event in S_(H)(y) or S_(H)(y+1) is represented asH_(kj), in which H_(k) is one or a set of variables indexed by k, forexample, H₁=BX₁, H₂=BX₂B₄, etc., and j indexes the state combination ofthe set of variables, for example, H_(1,2)=BX_(1,2),H_(2,3)=B_(1,3)X_(4,2), etc.

An example of DUCG is illustrated in FIG. 1. In DUCG, B-type variable orevent is drawn as rectangle, X-type variable or event is drawn ascircle, BX-type variable or event is drawn as double line circle, G-typevariable or event representing the logic relationship is drawn as gate,and D-type variable or event is drawn as pentagon representing thedefault or unknown cause of X-type variable/event. The {B−, X−, BX−, G−,D−}-type variable/events are also called nodes.

Variable G_(i) represents the logic combinations of input variables. Itmust have at least two inputs connected with directed arc

. The logic combinations are specified by logic gate specificationLGS_(i). For example, in FIG. 1 G₁ is specified by LGS₁:G_(11,1)=B_(3,1)X_(111,1), G_(11,2)=B_(3,1)X_(111,2), G_(1,0)=RemnantState that is defined as all other state combinations.

The textual descriptions of {B−, X−, BX−, D−, G−}-type variables andtheir states can be given according to their physical meanings. All {B−,X−, BX−, D−, G−}-type variable/event can be direct cause variable/event,and is called parent variable/event, and can be represented by V∈{B, X,BX, D, G} with the same index. For example, V₂=X₂, V_(3,2)=B_(3,2), etc.Direct consequence or child variables/events are usually {X−, BX−}-typevariables/events.

Directed arc

from parent to child is used to denote the weighted functional variableF_(n;i) representing the causal relationship between parent variableV_(i) and child variable X_(n) or BX_(n). F_(n;i) is an event matrix.F_(nk;ij)≡(r_(n;i)/r_(n))A_(nk;ij) represents the causal relationshipbetween parent event V_(ij) and child event X_(nk) or BX_(nk). Wherer_(n;i)>0 quantifies the uncertain causal relationship intensity betweenV_(i) and X_(n) or BX_(n), r_(n)≡Σ_(i)r_(n;i), and A_(nk;ij) representsthe uncertain causal mechanism that V_(ij) may cause X_(nk) or BX_(nk).The probability of A_(nk;ij) is defined as a_(nk;ij)=Pr{A_(nk;ij)}usually given by domain experts. Definef_(nk;ij)=Pr{F_(nk;ij)}=(r_(nk;ij)/r_(n))a_(nk;ij), in which f_(nk;ij)means the probabilistic contributions from V_(ij) to X_(nk), satisfying

${Pr\left\{ X_{nk} \right\}} = {\sum\limits_{i,j}{f_{{nk};{ij}}\Pr{\left\{ V_{ij} \right\}.}}}$F_(nk;ij), f_(nk;ij), A_(nk;ij) and a_(nk;ij) are member event inF_(n;i), f_(n;i), A_(n;i) and a_(n;i) respectively. Definev_(ij)=Pr{V_(ij)}, v∈{b, x, bx, d, g}. In general, the lower caseletters represent the probabilities of the corresponding upper caseletters that represent events or event variables.

The weighted functional event matrix F_(n;i) can also be conditional ona condition event Z_(n;i), which can be drawn as dashed directed arc

. Conditional F_(n;i) is used to represent the conditional causalrelationship between its parent event vector V_(i) and its child eventvector X_(nk) or BX_(nk). The condition event Z_(n;i) determines whetherF_(n;i) holds or not (discarded). Taken Z_(n;i)=X_(1,2) as an example,when X_(1,2) is true, F_(n;i) is held, i.e.

becomes

; when X_(1,2) is false, F_(n;i) is not held, i.e.

is discarded.

For simplicity, the letter of a variable in each graphical node symbolcan be ignored and only the indices are inside as shown in FIG. 2, inwhich the first number is the index of the variable and the secondnumber is the state index of the variable. Since D-type variable hasonly one state, there is only one index inside the symbol. Non-D-typevariables with known state can be indicated by colors, e.g. X_(110,1) isblue in FIG. 2. If there is only one number in the symbol, the state ofthe variable is unknown.

With the received evidence E(y)=E′(y)E″(y), where E′(y) is composed ofstate abnormal events and E″(y) is composed of state normal events, thefollowing rules can be used to simplify a DUCG:

Rule 1: If E(y) shows that Z_(n;i) is false, F_(n;i), is eliminated; ifE(y) shows that Z_(n;i) is true, the conditional F_(n;i) becomes theordinary F_(n;i), i.e.

becomes

3.

Rule 2: If E(y) shows that V_(ij) (V∈{B, X}) is true while V_(ij) is nota parent event of X_(n) or BX_(n), directed arc F_(n;i) is eliminated.

Rule 3: If E(y) shows that X_(nk) is true while X_(nk) cannot be causedby any state of V_(i) (V∈{B, X, BX, G, D}), directed arc F_(n;i) iseliminated.

Rule 4: If E(y) shows that a state-unknown X-type node does not have anyoutput and E(y) blocks its connection with B-type nodes, eliminate thisX-type node.

Rule 5: If E(y) shows X_(n0) is true, and X_(n0) has no connection withB-type nodes unless through the nodes in E(y), eliminate X_(n0).

Rule 6: If E(y) shows a group of state-unknown nodes that have noconnection with E′(y), this group of state-unknown nodes are eliminated.

Rule 7: Given E(y), if G_(i) has no output, eliminate G_(i) and itsinput directed arc

; if G_(i) has no input, eliminate G_(i) and its output directed arcs.

Rule 8: Given E(y), if a directed arc has no parent node or child node,eliminate this directed arc.

Rule 9: If there is a group of nodes and directed arcs that have noconnection with E′(y), this group of nodes and directed arcs areeliminated.

Rule 10: If E(y) shows X_(nk) (k≠0) is true while X_(nk) does not haveany input, add a virtual parent event D_(n) to X_(nk) as its input witha_(nk;nD)=1 and a_(n′;nD)=0 (k≠k′), where r_(n;D) can be any value. Theadded virtual D_(n) can be drawn as

Rule 11: Apply Rule 1-Rule 10 in any order, separately or together, andrepeatedly.

After applying the above rules, the DUCG is simplified, in whichH_(kj)=B_(kj) or H_(kj)=BX_(kj) (j≠0) compose S_(H)(y).

In the simplified DUCG, some state-unknown X-type variables arecause-specific, which means that the state observation of acause-specific X-type variable can determine whether the correspondinghypothesis event H_(kj) (H_(kj)∈S_(H)(y)) is true.

Technical references for this invention

-   [1] Q. Zhang and Z. Zhang. Method for constructing an intelligent    system processing uncertain causal relationship information. Chinese    patent: CN 200680055266.X, 2010.-   [2] Q. Zhang and Z. Zhang. Method for constructing an intelligent    system processing uncertain causal relationship information. U.S.    Pat. No. 8,255,353 B2, 2012.-   [3] Q. Zhang and C. Dong. Method for constructing cubic DUCG for    dynamic fault diagnosis. Chinese patent: CN 2013107185964, 2015.-   [4] Q. Zhang. “Dynamic uncertain causality graph for knowledge    representation and reasoning: discrete DAG cases”, Journal of    Computer Science and Technology, vol. 27, no. 1, pp. 1-23, 2012.-   [5] Q. Zhang, C. Dong, Y. Cui and Z. Yang. “Dynamic uncertain    causality graph for knowledge representation and probabilistic    reasoning: statistics base, matrix and fault diagnosis”, IEEE Trans.    Neural Networks and Learning Systems, vol. 25, no. 4, pp. 645-663,    2014.-   [6] Q. Zhang. “Dynamic uncertain causality graph for knowledge    representation and probabilistic reasoning: directed cyclic graph    and joint probability distribution”, IEEE Trans. Neural Networks and    Learning Systems, vol. 26, no. 7, pp. 1503-1517, 2015.-   [7] Q. Zhang. “Dynamic uncertain causality graph for knowledge    representation and probabilistic reasoning: continuous variable,    uncertain evidence and failure forecast”, IEEE Trans. Systems, Man    and Cybernetics, vol. 45, no. 7, pp. 990-1003, 2015.-   [8] Q. Zhang and S. Geng. “Dynamic uncertain causality graph applied    to dynamic fault diagnosis of large and complex systems”, IEEE    Trans. Reliability, vol. 64, no. 3, pp 910-927, 2015.-   [9] Q. Zhang and Z. Zhang. “Dynamic uncertain causality graph    applied to dynamic fault diagnoses and predictions with negative    feedbacks”, IEEE Trans. Reliability, vol. 65, no. 2, pp 1030-1044,    2016.

SUMMARY OF THE INVENTION

This invention discloses a technical scheme to rank the state-unknownX-type variables optimally, so that people can optically choose thestate-unknown X-type variables to detect (state pending X-typevariables) to obtain E⁺(y). Conditional on E(y+1)=E⁺(y)E(y), the moreaccurate S_(H)(y+1) and more accurate Pr{H_(kj)}, H_(kj)∈S_(H)(y+1), canbe found as earlier and with less cost as possible, while the dangerdegrees of H_(kj) are considered.

This invention is a subsequent invention and a further development ofgranted patents CN 200680055266.X, CN 2013107185964, and U.S. Pat. No.8,255,353 B2. The technical scheme of this invention comprises:

1. The method to rank the state pending X-type variables with at leastone CPU, according to the rank, part of or all of the state pendingX-type variables' states which form E⁺(y) are detected sequentially orparallel, in order to find the real cause H_(kj) that is in S_(H)(y+1)and to rank the real H_(kj) as high as possible conditioned onE(y+1)=E⁺(y)E(y), detailed steps include: (1) determine the detectablestate pending X-type variables whose index set is denoted as S_(X)(y)based on the simplified DUCG conditioned on E(y); (2) the ranking endsif S_(X)(y) contains only one element; (3) calculate the rank importanceI_(i)(y) of X_(i); (4) rank X_(i) (i∈S_(X)(y)) according to I_(i)(y) anddetect the states of X_(i) (i∈S_(X)(y)) in reference to the rank; (5) ifthe ranking is still needing, increase y to y+1, and repeat the abovestep (1)-(5) until the diagnosis is satisfied or no state pending X-typevariable available.

2. The method according to 1, wherein to determine the state pendingX-type variables with at least one CPU, the detailed steps include: (1)Collect all possible H_(kj) based on the simplified DUCG conditioned onE(y), these H_(kj) form S_(H)(y); (2) For each H_(k) in S_(H)(y), searchfor the state pending X-type variable connected to H_(k) withoutstate-known variables blocking them, where the indices of such X-typevariables make up S_(X)(y).

3. The method according to 1, wherein to determine the structureimportance λ_(i)(y)>0 of X_(i) for calculating I(y) with at least oneCPU, characterized in that: for the state pending X-type variables inthe above 1(1), count the number of its connected different H_(k)(k∈S_(iK)) in S_(H)(y) determined in the above 2, the number is writtenas m_(i)(y), calculate λ_(i)(y) based on m_(i)(y) according to a methodthat features at that the bigger m_(i)(y) is, the smaller λ_(i)(y) is,such method includes but not limits to λ_(i)(y)=1/(m_(i)(y))^(n) (n=1,2, . . . ).

4. The method according to 1, wherein assign a value to the dangerimportance ω_(kj)>0 of H_(kj) for calculating I_(i)(y) with at least oneCPU, characterized in that:

For each possible cause event H_(kj) in S_(H)(y), score all of theabnormal states according to their degree of concern, the score iscalled concern importance which is written as ω_(k), 1≥ω_(k)>0. Thegreater the concern is, the bigger ω_(k) is. The value of ω_(k) can beassigned when constructing DUCG or be assigned according to the concretesituation given H_(k) when S_(H)(y) is known.

5. The method according to 1, wherein to calculate the probabilityimportance ρ_(i)(y) for calculating I(y) with at least one CPU,characterized in that: calculate the average variation of theconditional probabilities of H_(k) in S_(H)(y) between the conditionsE(y) and X_(ig)E(y) over all abnormal states of all H_(k) connected withX without state-known variables blocking them, where g∈S_(iG)(y),S_(iG)(y) is the index set of possible states of X_(i), and X_(i) isnon-cause-specific state pending X-type variables, based on thesimplified DUCG conditioned on E(y), featuring at that the greater theaverage variation is, the bigger ρ_(i)(y) is, such as but are notlimited to:

${\rho_{i}(y)} = \left. {\frac{1}{m_{i}(y)}{\sum\limits_{k \in {S_{iK}{(y)}}}{\omega_{k}{\sum\limits_{j \in {S_{kJ}{(y)}}}{\sum\limits_{g \in {S_{iG}{(y)}}}{Pr\left\{ X_{ig} \middle| {E(y)} \right\}}}}}}} \middle| {{\Pr\left\{ H_{kj} \middle| {X_{ig}{E(y)}} \right\}} - {Pr\left\{ H_{kj} \middle| {E(y)} \right\}}} \middle| {Or} \right.$${\rho_{i}(y)} = {\frac{1}{m_{i}(y)}{\sum\limits_{k \in {S_{iK}{(y)}}}{\omega_{k}\sqrt{\sum\limits_{j \in {S_{kJ}{(y)}}}{\sum\limits_{g \in {S_{iG}{(y)}}}\left( {Pr\left\{ X_{ig} \middle| {E(y)} \right\}\left( {{Pr\left\{ H_{kj} \middle| {X_{ig}{E(y)}} \right\}} - {Pr\left\{ H_{kj} \middle| {E(y)} \right\}}} \right)} \right)^{2}}}}}}$Or${\rho_{i}(y)} = \left. {\frac{1}{m_{i}(y)}{\sum\limits_{k \in {S_{iK}{(y)}}}{\omega_{k}{\sum\limits_{j \in {S_{kJ}{(y)}}}\sum\limits_{g \in {S_{iG}{(y)}}}}}}} \middle| {{\Pr\left\{ H_{kj} \middle| {X_{ig}{E(y)}} \right\}} - {Pr\left\{ H_{kj} \middle| {E(y)} \right\}}} \middle| {Or} \right.$${\rho_{i}(y)} = {\frac{1}{m_{i}(y)}{\sum\limits_{k \in {S_{iK}{(y)}}}{\omega_{k}\sqrt{\sum\limits_{j \in {S_{kJ}{(y)}}}{\sum\limits_{g \in {S_{iG}{(y)}}}\left( {{Pr\left\{ H_{kj} \middle| {X_{ig}{E(y)}} \right\}} - {Pr\left\{ H_{kj} \middle| {E(y)} \right\}}} \right)^{2}}}}}}$Or${\rho_{i}(y)} = \left. {\frac{1}{m_{i}(y)}{\sum\limits_{k \in {S_{iK}{(y)}}}\omega_{k}}} \middle| {{\sum\limits_{g \in {S_{iG}{(y)}}}{Pr\left\{ H_{kj} \middle| {X_{ig}{E(y)}} \right\}}} - {\frac{1}{m_{i}}{\sum\limits_{{k \in {S_{iK}{(y)}}},{j \in {S_{kJ}{(y)}}}}{\sum\limits_{g \in {S_{iG}{(y)}}}{Pr\left\{ H_{kj} \middle| {X_{ig}{E(y)}} \right\}}}}}} \right|$E(y) and ω_(k) can be ignored, equivalent to being removed separately ortogether, that is to let E (y)=the complete set and ω_(k)=1.

6. The method according to 1 for calculating I(y) with at least one CPU,wherein to determine the cost importance, characterized in that:comprehensively assign a cost score for the state pending X_(i)(i∈S_(X)(y)), or calculate the cost score as the sum of the weightedscores assigned for the difficulty of doing the detection (indexed byj=1), waiting time (indexed by j=2), price (indexed by j=3) and damageto target system (indexed by j=4) respectively, the weights σ_(ij) canbe given when constructing DUCG or be given in an individualapplication, the bigger the cost score is, the smaller the costimportance β_(i) (1≥β_(i)>0) is, which including but not limited to:when the highest cost score is 100, β_(i)= 1/100=0.01, when the lowestcost score is 1, β_(i)=1/1=1, and the remnant values are between thehighest and the lowest case, can be assigned when constructing DUCG orbe given online for the state pending X-type variables included in thesimplified DUCG conditioned on E(y).

7. The method according to 5 to determine the probability importanceρ_(i)(y) with at least one CPU, characterized in that: based on thesimplified DUCG conditioned on E(y), for each cause-specific statepending X-type variable connected to H_(kj) (H_(kj) ∈S_(k)(y)), λ_(i)(y)and ρ_(i)(y) are not calculated according to the above 3 and 5, whichmeans S_(Xs)(y) is obtained by subtracting S_(s)(y) from S_(X)(y), butare calculated by selecting the maximum of i∈S_(s)(y), which includesbut is not limited to λ_(i)(y)≥1 and

${{\rho_{i}(y)} \geq {\max\limits_{l \in {S_{Xs}{(y)}}}\left\{ {\rho_{l}(y)} \right\}}},$that means X_(i) in the above 5 is limited to i∈S_(Xs)(y).

8. The method according to 1 to comprehensively calculate I(y) with atleast one CPU, characterized in that: the bigger λ_(i)(y) or ρ_(i)(y) orβ_(i) in the above 2-7 is, the bigger I_(i)(y) is, the detailedcalculation formulas include but are not limited to:

${I_{i}(y)} = \frac{{\lambda_{i}(y)}\beta_{i}{\rho_{i}(y)}}{\sum_{i \in {S_{X}{(y)}}}{{\lambda_{i}(y)}\beta_{i}{\rho_{i}(y)}}}$Or I_(i)(y) = λ_(i)(y)β_(i)ρ_(i)(y) Or${I_{i}(y)} = \frac{{\lambda_{i}(y)}{\rho_{i}(y)}}{\sum_{i \in {S_{X}{(y)}}}{{\lambda_{i}(y)}{\rho_{i}(y)}}}$Or I_(i)(y) = λ_(i)(y)ρ_(i)(y) Or${I_{i}(y)} = \frac{\beta_{i}{\rho_{i}(y)}}{\sum_{i \in {S_{X}{(y)}}}{\beta_{i}{\rho_{i}(y)}}}$Or I_(i)(y) = β_(i)ρ_(i)(y) Or${I_{i}(y)} = \frac{\rho_{i}(y)}{\sum_{i \in {S_{X}{(y)}}}{\rho_{i}(y)}}$Or I_(i)(y) = ρ_(i)(y) Or I_(i)(y) = w₁λ_(i)(y) + w₂ρ_(i)(y) + w₃β_(i)Or${I_{i}(y)} = \frac{{w_{1}{\lambda_{i}(y)}} + {w_{2}{\rho_{i}(y)}} + {w_{3}\beta_{i}}}{{\sum_{i \in {S_{X}{(y)}}}{w_{1}{\lambda_{i}(y)}}} + {w_{2}{\rho_{i}(y)}} + {w_{3}\beta_{i}}}$where w₁, w₂, and w₃ are three weights, and w_(i)≥0 (i=1, 2, 3), andw_(i)=0 means this item is not considered, wherein the value of w₁, w₂,and w₃ can be assigned when constructing DUCG or be assigned accordingto the individual application situation.

9. The method according to 1 to rank the state pending X-type variablesX_(i) (i∈S_(X)(y)) according to I_(i)(y) with at least one CPU,characterized in that: in the ranking, when the top-ranking X-typevariable is the only ancestor or descendant variable of the rankinglower X-type variable, the ranking lower X-type variable is eliminatedfrom the current rank, also the X_(i) whose I_(i)(y)=0 is eliminatedfrom the rank.

10. The method according to 1 to determine whether the ranking ends ornot with at least one CPU, characterized in that: the ranking ends ifthere is only one hypothesis event in S_(H)(y), or if all the statepending X-type variables are state-known, or there is only one elementin S_(X)(y).

The chart of the above steps is as shown in FIG. 3.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1: Illustration of DUCG.

FIG. 2: Illustration of a simple representation of DUCG.

FIG. 3: Step chart of this invention.

FIG. 4: The original DUCG of the examples.

FIG. 5: The results of dividing and simplifying FIG. 4 based on BX₁ wheny=0.

FIG. 6: The results of dividing and simplifying FIG. 4 based on BX₂ wheny=0.

FIG. 7: The results of dividing and simplifying FIG. 4 based on BX₃ wheny=0.

FIG. 8: The DUCG graph of example 1 after detecting the X-type variablesranking first 5.

FIG. 9: The simplified DUCG when y=1.

FIG. 10: The results of dividing and simplifying FIG. 9 based on BX₁when y=1.

FIG. 11: The results of dividing and simplifying FIG. 9 based on BX₂when y=2.

FIG. 12: The DUCG graph after detecting the states of X₈ and X₁₀.

FIG. 13: The simplified DUCG graph given E(2).

FIG. 14: The results of dividing and simplifying FIG. 13 based on BX₁.

FIG. 15: The results of dividing and simplifying FIG. 13 based on BX₂.

EXAMPLES TO IMPLEMENT THIS INVENTION

Suppose FIG. 4 is the original DUCG graph whose parameters are asfollows:

${b_{1} = \begin{pmatrix} - & 0.08\end{pmatrix}^{T}};{b_{2} = \begin{pmatrix} - & 0.01 & 0.02\end{pmatrix}^{T}};{b_{3} = \begin{pmatrix} - & 0.05\end{pmatrix}^{T}};{a_{1;1} = \begin{pmatrix} - & - \\ - & 1\end{pmatrix}};$ ${a_{2;2} = \begin{pmatrix} - & - & - \\ - & 1 & - \\ - & - & 1\end{pmatrix}};{a_{3;3} = \begin{pmatrix} - & - \\ - & 1\end{pmatrix}};{a_{12;D} = \begin{pmatrix} - \\0.1\end{pmatrix}};{a_{13;D} = \begin{pmatrix} - \\0.01\end{pmatrix}};{a_{14;D} = \begin{pmatrix} - \\0.1\end{pmatrix}};$ ${a_{1;12} = \begin{pmatrix} - & - \\ - & 0.7\end{pmatrix}};{a_{1;13} = \begin{pmatrix} - & - \\ - & 0.8\end{pmatrix}};{a_{3;14} = \begin{pmatrix} - & - \\ - & 0.6\end{pmatrix}};{a_{4;1} = \begin{pmatrix} - & - \\ - & 0.9\end{pmatrix}};$ ${a_{4;2} = \left( {\begin{matrix} - \\ - \end{matrix}\begin{matrix} - & - \\0.5 & 0.5\end{matrix}} \right)};{a_{5;1} = \begin{pmatrix} - & - \\ - & 0.7\end{pmatrix}};{a_{5;2} = \left( {\begin{matrix} - \\ - \end{matrix}\begin{matrix} - & - \\0.4 & 0.8\end{matrix}} \right)};{a_{5;3} = \begin{pmatrix} - & - \\ - & 0.7\end{pmatrix}};$ ${a_{6;2} = \begin{pmatrix} - & - & - \\ - & 0.6 & 0.3 \\ - & 0.4 & 0.7\end{pmatrix}};{a_{6;3} = \begin{pmatrix} - & - \\ - & 0.2 \\ - & 0.8\end{pmatrix}};{a_{7;4} = \begin{pmatrix} - & - \\ - & 0.7\end{pmatrix}};{a_{8;4} = \begin{pmatrix} - & - \\ - & 0.8\end{pmatrix}};$ ${a_{8;5} = \begin{pmatrix} - & - \\ - & 0.7\end{pmatrix}};{a_{8;6} = \begin{pmatrix} - & - & - \\ - & 0.2 & 0.8\end{pmatrix}};{a_{9;4} = \begin{pmatrix} - & - \\ - & 0.7\end{pmatrix}};{a_{10;5} = \begin{pmatrix} - & - \\ - & 0.7\end{pmatrix}};$ ${a_{11;6} = \begin{pmatrix} - & - & - \\ - & 0.3 & 0.8\end{pmatrix}};{a_{15;3} = \begin{pmatrix} - & - \\ - & 1\end{pmatrix}};{a_{16;3} = \begin{pmatrix} - & - \\ - & 1\end{pmatrix}};$r_(n;i)=1 are assumed. Among all the X-type variables, X₁₄ is anon-detectable variable; other X-type variables are all detectable. X₁₅and X₁₆ are cause-specific variables of BX_(3,1), that is when X_(15,1)or X_(16,1) is true, BX_(3,1) must be true. Since {B_(1,1), B_(2,1),B_(2,2), B_(3,1)} is equivalent to {BX_(1,1), BX_(2,1), BX_(2,2),BX_(3,1)}, the B-type variables are not treated as the diagnosis object.In other words, H-type events are composed of only BX-type events.

The parameter of the directed arc between X_(n) and D_(n) can be brieflywritten as a_(n;D). Our task is to detect the states of the statepending X-type variables as few and with less cost as possible, in orderto minimize the size of the possible set of cause events S_(H)(y) and tomaximize the probability of real cause event.

Example 1: y=0 (with No Evidence)

According to 2, when there is no evidence (y=0), based on FIG. 4,S_(H)(0)={H_(1,1), H_(2,1), H_(2,2), H_(3,1)}={BX_(1,1), BX_(2,1),BX_(2,2), BX_(3,1)}, and the state pending X-type variables are {X₄, X₅,X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₅, X₁₆}. All these states areunknown and single-connect to H_(k) in S_(H)(0) without state-knownblocking variables, so that S_(X)(0)={4, 5, 6, 7, 8, 9, 10, 11, 12, 13,15, 16}. Since X₁₅ and X₁₆ are cause-specific variables of BX_(3,1),they are eliminated from S_(X)(0), that means S_(Xs)(0)={4, 5, 6, 7, 8,9, 10, 11, 12, 13}. Accordingly,S_(4G)(0)=S_(5G)(0)=S_(7G)(0)=S_(8G)(0)=S_(9G)(0)=S_(10G)(0)=S_(11G)(0)=S_(12G)(0)=S_(13G)(0)={1}and S_(6G)(0)={1, 2}.

Based on FIG. 4, according to the DUCG algorithms in Ref [8], since theintersection of different H_(kj) is null set, which means differentH_(kj) cannot occur simultaneously, FIG. 4 can be divided according toH_(k) and can be simplified according to simplification rules, theresults are shown in FIG. 5-FIG. 7.

Since there is no evidence, E(0)=1 (complete set), the probability andweighting factor of each sub-graph are ζ_(i)(y)=ζ_(i)(0)=Pr{E(0)}=1(i∈{1,2,3}) and

${\xi_{i}(y)} = {{\xi_{i}(0)} = {{{\zeta_{i}(0)}/{\sum\limits_{i}{\zeta_{i}(0)}}} = {{\frac{1}{3}{\xi_{i}(y)}} = {{\xi_{i}(0)} = {{{\zeta_{i}(0)}/{\sum\limits_{i}{\zeta_{i}(0)}}} = {\frac{1}{3}.}}}}}}$Based on FIG. 5-FIG. 7, according to algorithms in the references, thestate probabilities h_(kj) ^(s)(y)=h_(kj) ^(s)(0) of BX_(1,1), BX_(2,1),BX_(2,2) and BX_(3,1) are:

$\begin{matrix}{{h_{1,1}^{s}(0)} = {{\xi_{1}(0)}\Pr\left\{ {{BX}_{1,1}❘{E(0)}} \right\}}} \\{= {{\xi_{1}(0)}\Pr\left\{ {BX}_{1,1} \right\}}} \\{= {{\xi_{1}(0)}\Pr\left\{ {{\frac{r_{1;1}}{r_{1}}A_{1,{1;1}}B_{1}} + {\frac{r_{1;12}}{r_{1}}A_{1,{1;12}}A_{12;D}} + {\frac{r_{1;13}}{r_{1}}A_{1,{1;13}}A_{13;D}}} \right\}}} \\{= {{\frac{1}{3} \times \frac{1}{3}a_{1,{1;1}}b_{1}} + {\frac{1}{3}a_{1,{1;12}}a_{12;D}} + {\frac{1}{3}a_{1,{1;13}}a_{13;D}}}} \\{= {{\frac{1}{3} \times \frac{1}{3}\begin{pmatrix} - & 1\end{pmatrix}\begin{pmatrix} - \\0.08\end{pmatrix}} + {\frac{1}{3}\begin{pmatrix} - & 0.7\end{pmatrix}\begin{pmatrix} - \\0.1\end{pmatrix}} + {\frac{1}{3}\begin{pmatrix} - & 0.8\end{pmatrix}\begin{pmatrix} - \\0.01\end{pmatrix}}}} \\{= 0.01756}\end{matrix}$ $\begin{matrix}{{h_{2,1}^{s}(0)} = {{\xi_{2}(0)}\Pr\left\{ {BX}_{2,1} \right\}}} \\{= {{\xi_{2}(0)}\Pr\left\{ {A_{2,{1;2}}B_{2}} \right\}}} \\{= {{\xi_{2}(0)}a_{2,{1;2}}b_{2}}} \\{= {\frac{1}{3}\begin{pmatrix} - & 1 & - \end{pmatrix}\begin{pmatrix} - & 0.01 & 0.02\end{pmatrix}^{T}}} \\{= 0.003333}\end{matrix}$ $\begin{matrix}{{h_{2,2}^{s}(0)} = {{\xi_{2}(0)}\Pr\left\{ {BX}_{2,2} \right\}}} \\{= {{\xi_{2}(0)}\Pr\left\{ {A_{2,{2;2}}B_{2}} \right\}}} \\{= {{\xi_{2}(0)}a_{2,{2;2}}b_{2}}} \\{= {\frac{1}{3}\begin{pmatrix} - & - & 1\end{pmatrix}\begin{pmatrix} - & 0.01 & 0.02\end{pmatrix}^{T}}} \\{= 0.006667}\end{matrix}$ $\begin{matrix}{{h_{3,1}^{s}(0)} = {{\xi_{3}(0)}\Pr\left\{ {BX}_{3,1} \right\}}} \\{= {{\xi_{3}(0)}\Pr\left\{ {{\frac{r_{3;3}}{r_{3}}A_{3,{1;3}}B_{3}} + {\frac{r_{3;14}}{r_{3}}A_{3,{1;14}}A_{14;D}}} \right\}}} \\{= {{\frac{1}{3} \times \frac{1}{2}a_{3,{1;3}}b_{3}} + {\frac{1}{2}a_{3,{1;14}}a_{14;D}}}} \\{= {{\frac{1}{3} \times \frac{1}{2}\begin{pmatrix} - & 1\end{pmatrix}\begin{pmatrix} - \\0.05\end{pmatrix}} + {\frac{1}{2}\begin{pmatrix} - & 0.6\end{pmatrix}\begin{pmatrix} - \\0.1\end{pmatrix}}}} \\{= 0.01833}\end{matrix}$The calculation result of rank probability

${h_{kj}^{r}(y)} = \frac{h_{kj}^{s}(y)}{\sum\limits_{H_{kj} \in {S_{H}{(y)}}}{h_{kj}^{s}(y)}}$is as follows:

index H_(kj) h_(kj) ^(r) (0) 1 BX_(3, 1) 0.3995 2 BX_(1, 1) 0.3826 3BX_(2, 2) 0.1453 4 BX_(2, 1) 0.0725

According to 3, based on FIG. 4, one can get S_(12K)(0)=S_(13K)(0)={1},and S_(4K)(0)S_(7K)(0)S_(9K)(0)={1,2}, and S_(6K)(0)=S_(11K)(0)={2,3},and S_(5K)(0)=S_(8K)(0)=S_(10K)(0)={1,2,3}. Accordingly, one can getm₁₂(0)=m₁₃(0)=1, and m₄(0)=m₆(0)=m₇(0)=m₉(0)=m₁₁(0)=2, andm₅(0)=m₈(0)=m₁₀(0)=3. Since BX₁ and BX₃ have only one abnormal stateindexed by “1”, thus S_(1J)(0)=S_(3J)(0)={1}. And BX₂ has two abnormalstates indexed by “1” and “2”, thus S_(2J)(0)={1,2}.

According to 4, assume ω_(kj)=1 (k∈{1,2,3}).

According to 5, with the following formula:

${\rho_{i}(y)} = \left. {\frac{1}{m_{i}^{\prime}(y)}{\sum\limits_{k \in {S_{iK}{(y)}}}{\sum\limits_{j \in {S_{kJ}{(y)}}}{\omega_{kj}{\sum\limits_{g \in {S_{iG}{(y)}}}{Pr\left\{ X_{ig} \middle| {E(y)} \right\}}}}}}} \middle| {{\Pr\left\{ H_{kj} \middle| {X_{ig}{E(y)}} \right\}} - {Pr\left\{ H_{kj} \middle| {E(y)} \right\}}} \right|$where i∈S_(Xs)(0)={4, 5, 6, 7, 8, 9, 10, 11, 12, 13} andm′_(i)(y)=m_(i)(y). Based on FIG. 4, since ω_(kj)=1, k∈{1,2,3}, thus

$\begin{matrix}{{\rho_{4}(0)} = \left. {\frac{1}{m_{4}(0)}{\sum\limits_{k \in {\{{1,2}\}}}{\sum\limits_{j \in {S_{kJ}{(0)}}}{\sum\limits_{g \in {\{ 1\}}}{Pr\left\{ X_{ig} \middle| {E(0)} \right\}}}}}} \right|} \\{\left. {{\Pr\left\{ H_{kj} \middle| {X_{ig}{E(0)}} \right\}} - {Pr\left\{ H_{kj} \middle| {E(0)} \right\}}} \right|} \\{= {\frac{1}{2}Pr\left\{ X_{4,1} \right\}\begin{pmatrix}{{\left. {\Pr\left\{ H_{1,1} \right.X_{4,1}} \right\} - {\Pr\left\{ H_{1,1} \right\}}}} \\{{+ \left. {\Pr\left\{ H_{2,1} \right.X_{4,1}} \right\}} - {\Pr\left\{ H_{2,1} \right\}}} \\{+ {{{\Pr\left\{ {H_{2,2}❘X_{4,1}} \right\}} - {\Pr\left\{ H_{2,2} \right\}}}}}\end{pmatrix}}} \\{= {\frac{1}{2}Pr\left\{ X_{4,1} \right\}\begin{pmatrix}{{{\Pr\left\{ {{BX}_{1,1}❘X_{4,1}} \right\}} - {\Pr\left\{ {BX}_{1,1} \right\}}}} \\{+ {{{\Pr\left\{ {{BX}_{2,1}❘X_{4,1}} \right\}} - {\Pr\left\{ {BX}_{2,1} \right\}}}}} \\{+ {{{\Pr\left\{ {{BX}_{2,2}❘X_{4,1}} \right\}} - {\Pr\left\{ {BX}_{2,2} \right\}}}}}\end{pmatrix}}} \\{= {\frac{1}{2}\begin{pmatrix}{{{\Pr\left\{ {{BX}_{1,1}X_{4,1}} \right\}} - {\Pr\left\{ X_{4,1} \right\}\Pr\left\{ {BX}_{1,1} \right\}}}} \\{+ {{{\Pr\left\{ {{BX}_{2,1}X_{4,1}} \right\}} - {\Pr\left\{ X_{4,1} \right\}\Pr\left\{ {BX}_{2,1} \right\}}}}} \\{+ {{{\Pr\left\{ {{BX}_{2,2}X_{4,1}} \right\}} - {\Pr\left\{ X_{4,1} \right\}\Pr\left\{ {BX}_{2,2} \right\}}}}}\end{pmatrix}}} \\{= {\frac{1}{2}\begin{pmatrix}{\begin{matrix}{\Pr\left\{ {{BX}_{1,1}\left( {{\frac{r_{4;1}}{r_{4}}A_{4,{1;1}}{BX}_{1}} + {\frac{r_{4;2}}{r_{4}}A_{4,{1;2}}{BX}_{2}}} \right)} \right\}} \\{{- \Pr}\left\{ X_{4,1} \right\}\Pr\left\{ {BX}_{1,1} \right\}}\end{matrix}} \\{+ {\begin{matrix}{\Pr\left\{ {{BX}_{2,1}\left( {{\frac{r_{4;1}}{r_{4}}A_{4,{1;1}}{BX}_{1}} + {\frac{r_{4;2}}{r_{4}}A_{4,{1;2}}{BX}_{2}}} \right)} \right\}} \\{{- \Pr}\left\{ X_{4,1} \right\}\Pr\left\{ {BX}_{2,1} \right\}}\end{matrix}}} \\{+ {\begin{matrix}{\Pr\left\{ {{BX}_{2,2}\left( {{\frac{r_{4;1}}{r_{4}}A_{4,{1;1}}{BX}_{1}} + {\frac{r_{4;2}}{r_{4}}A_{4,{1;2}}{BX}_{2}}} \right)} \right\}} \\{{- \Pr}\left\{ X_{4,1} \right\}\Pr\left\{ {BX}_{2,2} \right\}}\end{matrix}}}\end{pmatrix}}} \\{= {\frac{1}{2}\begin{pmatrix}{{{\Pr\left\{ {A_{4,{1;1},1}{BX}_{1,1}} \right\}} - {\Pr\left\{ X_{4,1} \right\}\Pr\left\{ {BX}_{1,1} \right\}}}} \\{+ {{{\Pr\left\{ {A_{4,{1;2},1}{BX}_{2,1}} \right\}} - {\Pr\left\{ X_{4,1} \right\}\Pr\left\{ {BX}_{2,1} \right\}}}}} \\{+ {{{\Pr\left\{ {A_{4,{1;2},2}{BX}_{2,2}} \right\}} - {\Pr\left\{ X_{4,1} \right\}\Pr\left\{ {BX}_{2,2} \right\}}}}}\end{pmatrix}}} \\{= {\frac{1}{2}\begin{pmatrix}{{\left( {{\Pr\left\{ A_{4,{1;1},1} \right\}} - {\Pr\left\{ X_{4,1} \right\}}} \right)\Pr\left\{ {BX}_{1,1} \right\}}} \\{+ {{\left( {{\Pr\left\{ A_{4,{1;2},1} \right\}} - {\Pr\left\{ X_{4,1} \right\}}} \right)\Pr\left\{ {BX}_{2,1} \right\}}}} \\{+ {{\left( {{\Pr\left\{ A_{4,{1;2},2} \right\}} - {\Pr\left\{ X_{4,1} \right\}}} \right)\Pr\left\{ {BX}_{2,2} \right\}}}}\end{pmatrix}}}\end{matrix}$In which

$\begin{matrix}{{\Pr\left\{ X_{4.1} \right\}} = {\Pr\left\{ {{\frac{r_{4;1}}{r_{4}}{A_{4,{1;1}}\ \begin{pmatrix}{\frac{r_{1;1}}{r_{1}}A_{1;1}B_{1}} \\{{+ \frac{r_{1;12}}{r_{1}}}A_{1;12}A_{12;D}} \\{{+ \frac{r_{1;13}}{r_{1}}}A_{1;13}A_{13;D}}\end{pmatrix}}} + {\frac{r_{4;2}}{r_{4}}A_{4,{1;2}}BX_{2}}} \right\}}} \\{= {\Pr\left\{ {{\frac{1}{2}{A_{4,{1;1}}\ \begin{pmatrix}{\frac{1}{3}A_{1;1}B_{1}} \\{{+ \frac{1}{3}}A_{1;12}A_{12;D}} \\{{+ \frac{1}{3}}A_{1;13}A_{13;D}}\end{pmatrix}}} + {\frac{1}{2}A_{4,{1;2}}A_{2;2}B_{2}}} \right\}}} \\{= {{\frac{1}{2}{a_{4,{1;1}}\ \begin{pmatrix}{\frac{1}{3}a_{1;1}b_{1}} \\{{+ \frac{1}{3}}a_{1;12}a_{12;D}} \\{{+ \frac{1}{3}}a_{1;13}a_{13;D}}\end{pmatrix}}} + {\frac{1}{2}a_{4,{1;2}}a_{2;2}b_{2}}}} \\{= {{\frac{1}{2}\begin{pmatrix} - & 0.9\end{pmatrix}\begin{pmatrix}{\frac{1}{3}\begin{pmatrix} - & - \\ - & 1\end{pmatrix}\begin{pmatrix} - \\0.08\end{pmatrix}} \\{{+ \frac{1}{3}}\begin{pmatrix} - & - \\ - & 0.7\end{pmatrix}\begin{pmatrix} - \\0.1\end{pmatrix}} \\{{+ \frac{1}{3}}\begin{pmatrix} - & - \\ - & 0.8\end{pmatrix}\begin{pmatrix} - \\0.01\end{pmatrix}}\end{pmatrix}} + {\frac{1}{2}\begin{pmatrix} - & 0.5 & 0.5\end{pmatrix}}}} \\{\begin{pmatrix} - & - & - \\ - & 1 & - \\ - & - & 1\end{pmatrix}\begin{pmatrix} - \\0.01 \\0.02\end{pmatrix}} \\{= 0.0312}\end{matrix}$Thus:

$\begin{matrix}{{\rho_{4}(0)} = {\frac{1}{2}\begin{pmatrix}{{\left( {a_{4,{1;1},1} - {\Pr\left\{ X_{4,1} \right\}}} \right)\Pr\left\{ {BX}_{1,1} \right\}}} \\{+ {{\left( {a_{4,{1;2},1} - {\Pr\left\{ X_{4,1} \right\}}} \right)\Pr\left\{ {BX}_{2,1} \right\}}}} \\{+ {{\left( {a_{4,{1;2},2} - {\Pr\left\{ X_{4,1} \right\}}} \right)\Pr\left\{ {BX}_{2,2} \right\}}}}\end{pmatrix}}} \\{= {{\frac{1}{2}\begin{pmatrix}{{\left( {0.9 - 0.0312} \right)0.05267}} \\{+ {{\left( {0.5 - 0.0312} \right)0.01}}} \\{+ {{\left( {0.5 - 0.0312} \right)0.02}}}\end{pmatrix}} = 0.0299}}\end{matrix}$Similarly,

$\begin{matrix}{{\rho_{5}(0)} = {\frac{1}{m_{5}(0)}{\sum\limits_{k \in {\{{1,2,3}\}}}^{\;}\;{\sum\limits_{j \in {S_{kJ}{(0)}}}^{\;}\;{\sum\limits_{g \in {\{ 1\}}}^{\;}\;{\Pr\left\{ {X_{ig}\left. {E(0)} \right\}} \right.\Pr\left\{ {{H_{kj}\left. {X_{ig}{E(0)}} \right\}} -} \right.}}}}}} \\{\Pr\left\{ {H_{kj}\left. {E(0)} \right\}} \right.} \\{= {\frac{1}{3}\Pr\left\{ X_{5,1} \right\}\begin{pmatrix}{\left. {\Pr\left\{ {BX}_{1,1} \right.X_{5,1}} \right\} - {\Pr\left\{ {BX}_{1,1} \right\} }} \\\left. {{+ \left. {\Pr\left\{ {BX}_{2,1} \right.X_{5,1}} \right\}} - {\Pr\left\{ {BX}_{2,1} \right\}{ + }\Pr\left\{ {BX}_{2,2} \right.X_{5,1}}} \right\} \\{{- \Pr}\left\{ {BX}_{2,2} \right\}{{{+ \left. {\Pr\left\{ {BX}_{3,1} \right.X_{5,1}} \right\}} - {\Pr\left\{ {BX}_{3,1} \right\}}}}}\end{pmatrix}}} \\{= 0.030331}\end{matrix}$ $\begin{matrix}{{\rho_{6}(0)} = {\frac{1}{m_{6}(0)}{\sum\limits_{k \in {\{{2,3}\}}}^{\;}\;{\sum\limits_{j \in {S_{kJ}{(0)}}}^{\;}\;{\sum\limits_{g \in {\{{1,2}\}}}^{\;}\;{\Pr\left\{ {X_{ig}\left. {E(0)} \right\}} \right.\Pr\left\{ {{H_{kj}\left. {X_{ig}{E(0)}} \right\}} -} \right.}}}}}} \\{\Pr\left\{ {H_{kj}\left. {E(0)} \right\}} \right.} \\{= {\frac{1}{2}\Pr\left\{ X_{6,1} \right\}\begin{pmatrix}{\left. {\left. {\Pr\left\{ {BX}_{2,1} \right.X_{6,1}} \right\} - {\Pr\left\{ {BX}_{2,1} \right\}{ + }\Pr\left\{ {BX}_{2,2} \right.X_{6,1}}} \right\} -} \\{{{\Pr\left\{ {BX}_{2,2} \right\}}} + {{\Pr\left\{ {{{BX}_{3,1}\left. X_{6,1} \right\}} - {\Pr\left\{ {BX}_{3,1} \right\}}} \right.}}}\end{pmatrix}}} \\{{+ \frac{1}{2}}{\Pr\left( X_{6,2} \right\}}\begin{pmatrix}{\left. {\left. {\Pr\left\{ {BX}_{2,1} \right.X_{6,2}} \right\} - {\Pr\left\{ {BX}_{2,1} \right\}{ + }\Pr\left\{ {BX}_{2,2} \right.X_{6,2}}} \right\} -} \\{{\left. {{{{{\Pr\left\{ {BX}_{2,2} \right\}}} +}}\Pr\left\{ {BX}_{3,1} \right.X_{6,2}} \right\} - {\Pr\left\{ {BX}_{3,1} \right\}}}}\end{pmatrix}} \\{= 0.040694}\end{matrix}$ $\begin{matrix}{{\rho_{7}(0)} = {\frac{1}{m_{7}(0)}{\sum\limits_{k \in {\{{1,2}\}}}^{\;}\;{\sum\limits_{j \in {S_{kJ}{(0)}}}^{\;}\;{\sum\limits_{g \in {\{ 1\}}}^{\;}\;{\Pr\left\{ {X_{ig}\left. {E(0)} \right\}} \right.\Pr\left\{ {{H_{kj}\left. {X_{ig}{E(0)}} \right\}} -} \right.}}}}}} \\{\Pr\left\{ {H_{kj}\left. {E(0)} \right\}} \right.} \\{= {\frac{1}{2}\Pr\left\{ X_{7,1} \right\}\begin{pmatrix}{{\left. {\Pr\left\{ {BX}_{1,1} \right.X_{7,1}} \right\} - {\Pr\left\{ {BX}_{1,1} \right\}}}} \\{{{+ \left. {\Pr\left\{ {BX}_{2,1} \right.X_{7,1}} \right\}} - {\Pr\left\{ {BX}_{2,1} \right\}}}} \\{{{+ \left. {\Pr\left\{ {BX}_{2,2} \right.X_{7,1}} \right\}} - {\Pr\left\{ {BX}_{2,2} \right\}}}}\end{pmatrix}}} \\{= 0.020938}\end{matrix}$ $\begin{matrix}{{\rho_{8}(0)} = {\frac{1}{m_{8}(0)}{\sum\limits_{k \in {\{{1,2,3}\}}}^{\;}\;{\sum\limits_{j \in {S_{kJ}{(0)}}}^{\;}\;{\sum\limits_{g \in {\{ 1\}}}^{\;}\;{\Pr\left\{ {X_{ig}\left. {E(0)} \right\}} \right.\Pr\left\{ {{H_{kj}\left. {X_{ig}{E(0)}} \right\}} -} \right.}}}}}} \\{\Pr\left\{ {H_{kj}\left. {E(0)} \right\}} \right.} \\{= {\frac{1}{3}\Pr\left\{ X_{8,1} \right\}\begin{pmatrix}{\left. {\Pr\left\{ {BX}_{1,1} \right.X_{8,1}} \right\} - {\Pr\left\{ {BX}_{1,1} \right\} }} \\\left. {{+ \left. {\Pr\left\{ {BX}_{2,1} \right.X_{8,1}} \right\}} - {\Pr\left\{ {BX}_{2,1} \right\}{ + }\Pr\left\{ {BX}_{2,2} \right.X_{8,1}}} \right\} \\{{- \Pr}\left\{ {BX}_{2,2} \right\}{{{+ \left. {\Pr\left\{ {BX}_{3,1} \right.X_{8,1}} \right\}} - {\Pr\left\{ {BX}_{3,1} \right\}}}}}\end{pmatrix}}} \\{= 0.01785}\end{matrix}$ $\begin{matrix}{{\rho_{9}(0)} = {\frac{1}{m_{9}(0)}{\sum\limits_{k \in {\{{1,2}\}}}^{\;}\;{\sum\limits_{j \in {S_{kJ}{(0)}}}^{\;}\;{\sum\limits_{g \in {\{ 1\}}}^{\;}\;{\Pr\left\{ {X_{ig}\left. {E(0)} \right\}} \right.\Pr\left\{ {{H_{kj}\left. {X_{ig}{E(0)}} \right\}} -} \right.}}}}}} \\{\Pr\left\{ {H_{kj}\left. {E(0)} \right\}} \right.} \\{= {\frac{1}{2}\Pr\left\{ X_{9,1} \right\}\begin{pmatrix}{{\left. {\Pr\left\{ {BX}_{1,1} \right.X_{9,1}} \right\} - {\Pr\left\{ {BX}_{1,1} \right\}}}} \\{{{+ \left. {\Pr\left\{ {BX}_{2,1} \right.X_{9,1}} \right\}} - {\Pr\left\{ {BX}_{2,1} \right\}}}} \\{{{+ \left. {\Pr\left\{ {BX}_{2,2} \right.X_{9,1}} \right\}} - {\Pr\left\{ {BX}_{2,2} \right\}}}}\end{pmatrix}}} \\{= 0.020938}\end{matrix}$ $\begin{matrix}{{\rho_{10}(0)} = {\frac{1}{m_{10}(0)}{\sum\limits_{k \in {\{{1,2,3}\}}}^{\;}\;{\sum\limits_{j \in {S_{kJ}{(0)}}}^{\;}\;{\sum\limits_{g \in {\{ 1\}}}^{\;}\;{\Pr\left\{ {X_{ig}\left. {E(0)} \right\}} \right.\Pr\left\{ {{H_{kj}\left. {X_{ig}{E(0)}} \right\}} -} \right.}}}}}} \\{\Pr\left\{ {H_{kj}\left. {E(0)} \right\}} \right.} \\{= {\frac{1}{3}\Pr\left\{ X_{10,1} \right\}\begin{pmatrix}{\left. {\Pr\left\{ {BX}_{1,1} \right.X_{10,1}} \right\} - {\Pr\left\{ {BX}_{1,1} \right\} }} \\\left. {{+ \left. {\Pr\left\{ {BX}_{2,1} \right.X_{10,1}} \right\}} - {\Pr\left\{ {BX}_{2,1} \right\}{ + }\Pr\left\{ {BX}_{2,2} \right.X_{10,1}}} \right\} \\{{- \Pr}\left\{ {BX}_{2,2} \right\}{{{+ \left. {\Pr\left\{ {BX}_{3,1} \right.X_{10,1}} \right\}} - {\Pr\left\{ {BX}_{3,1} \right\}}}}}\end{pmatrix}}} \\{= 0.021232}\end{matrix}$ $\begin{matrix}{{\rho_{11}(0)} = {\frac{1}{m_{11}(0)}{\sum\limits_{k \in {\{{2,3}\}}}^{\;}\;{\sum\limits_{j \in {S_{kJ}{(0)}}}^{\;}\;{\sum\limits_{g \in {\{ 1\}}}^{\;}\;{\Pr\left\{ {X_{ig}\left. {E(0)} \right\}} \right.\Pr\left\{ {{H_{kj}\left. {X_{ig}{E(0)}} \right\}} -} \right.}}}}}} \\{\Pr\left\{ {H_{kj}\left. {E(0)} \right\}} \right.} \\{= {\frac{1}{2}\Pr\left\{ X_{11,1} \right\}\begin{pmatrix}{\left. {\left. {\Pr\left\{ {BX}_{2,1} \right.X_{11,1}} \right\} - {\Pr\left\{ {BX}_{2,1} \right\}{ + }\Pr\left\{ {BX}_{2,2} \right.X_{11,1}}} \right\} -} \\{{\left. {\Pr\left\{ {BX}_{2,2} \right\}{ + }\Pr\left\{ {BX}_{3,1} \right.X_{11,1}} \right\} - {\Pr\left\{ {BX}_{3,1} \right\}}}}\end{pmatrix}}} \\{= 0.027049}\end{matrix}$ $\begin{matrix}{{\rho_{12}(0)} = {\frac{1}{m_{12}(0)}{\sum\limits_{k \in {\{ 1\}}}^{\;}\;{\sum\limits_{j \in {\{ 1\}}}^{\;}\;{\sum\limits_{g \in {\{ 1\}}}^{\;}\;{\Pr\left\{ {X_{12g}\left. {E(0)} \right\}} \right.\Pr\left\{ {{H_{kj}\left. {X_{ig}{E(0)}} \right\}} -} \right.}}}}}} \\{\Pr\left\{ {H_{kj}\left. {E(0)} \right\}} \right.} \\{{= {{\frac{1}{1}\Pr\left\{ X_{12,1} \right\}\left. {\Pr\left\{ {BX}_{1,1} \right.X_{12,1}} \right\}} - {\Pr\left\{ {BX}_{1,1} \right\}}}}} \\{= 0.063}\end{matrix}$ $\begin{matrix}{{\rho_{13}(0)} = {\frac{1}{m_{13}(0)}{\sum\limits_{k \in {\{ 1\}}}^{\;}\;{\sum\limits_{j \in {\{ 1\}}}^{\;}\;{\sum\limits_{g \in {\{ 1\}}}^{\;}\;{\Pr\left\{ {X_{13g}\left. {E(0)} \right\}} \right.\Pr\left\{ {{H_{kj}\left. {X_{ig}{E(0)}} \right\}} -} \right.}}}}}} \\{\Pr\left\{ {H_{kj}\left. {E(0)} \right\}} \right.} \\{{= {{\frac{1}{1}\Pr\left\{ X_{13,1} \right\}\left. {\Pr\left\{ {BX}_{1,1} \right.X_{13,1}} \right\}} - {\Pr\left\{ {BX}_{1,1} \right\}}}}} \\{= 0.00792}\end{matrix}$

According to 7, take

${{\rho_{i}(y)} = {\max\limits_{l \in {S_{Xs}{(y)}}}\left\{ {\rho_{l}(y)} \right\}}},$one can get the following:ρ₁₅(0)=ρ₁₆(0)=max{0.0299, 0.030331, 0.040694, 0.020938, 0.01785,0.020838, 0.021232, 0.027049, 0.063, 0.00792}=0.063.And then:

i ρ_(i)(0) 4 0.0299 5 0.030331 6 0.040694 7 0.020938 8 0.01785 90.020938 10 0.021232 11 0.027049 12 0.063 13 0.00792 15 0.063 16 0.063According to 6, let β_(i)=1/α_(i) and assign their values as follows:

i α_(i) β_(i) 4 100 0.01 5 2 0.5 6 50 0.02 7 1 1 8 2 0.5 9 2 0.5 10 20.5 11 1 1 12 1 1 13 1 1 15 1 1 16 1 1According to 3, assume n=1, given m_(i)(0), λ_(i)(0) is calculated asfollows:

i m_(i)(0) λ_(i)(0) 4 2 1/2 5 3 1/3 6 2 1/2 7 1 1 8 3 1/3 9 1 1 10 1 111 1 1 12 1 1 13 1 1 15 1 1 16 1 1In which, according to 7, let λ₁₅(0)=λ₁₆(0)=1.According to 8, take

${{I_{i}(y)} = \frac{{\lambda_{i}(y)}\beta_{i}{\rho_{i}(y)}}{\Sigma_{i \in S_{X^{(y)}}}{\lambda_{i}(y)}\beta_{i}{\rho_{i}(y)}}},$we can obtain the following results:

${I_{4}(0)} = {\frac{{\lambda_{4}(0)}\beta_{4}{\rho_{4}(0)}}{\sum\limits_{i \in {\{{4,\ldots,13,15,16}\}}}{{\lambda_{i}(0)}\beta_{i}{\rho_{i}(0)}}} = {\frac{\frac{1}{2} \times 0.01 \times 0.0299}{{0.2}7457861} = {{0.0}00544}}}$${I_{5}(0)} = {\frac{{\lambda_{5}(0)}\beta_{5}{\rho_{5}(0)}}{\sum\limits_{i \in {\{{4,\ldots,13,15,16}\}}}{{\lambda_{i}(0)}\beta_{i}{\rho_{i}(0)}}} = {\frac{\frac{1}{3} \times {0.5} \times {0.0}30331}{{0.2}7457861} = {{0.0}18411}}}$${I_{6}(0)} = {\frac{{\lambda_{6}(0)}\beta_{6}{\rho_{6}(0)}}{\sum\limits_{i \in {\{{{4\text{,…,}13},15,16}\}}}{{\lambda_{i}(0)}\beta_{i}{\rho_{i}(0)}}} = {\frac{\frac{1}{2} \times {0.0}2 \times 0.040694}{{0.2}7457861} = {{0.0}01482}}}$${I_{7}(0)} = {\frac{{\lambda_{7}(0)}\beta_{7}{\rho_{7}(0)}}{\sum\limits_{i \in {\{{4,{\text{...,}13},15,16}\}}}{{\lambda_{i}(0)}\beta_{i}{\rho_{i}(0)}}} = {\frac{1 \times 1 \times 0.020938}{{0.2}7457861} = {{0.0}76255}}}$${I_{8}(0)} = {\frac{{\lambda_{8}(0)}\beta_{8}{\rho_{8}(0)}}{\sum\limits_{i \in {\{{4,{\text{...,}13},15,16}\}}}{{\lambda_{i}(0)}\beta_{i}{\rho_{i}(0)}}} = {\frac{\frac{1}{3} \times 0.5 \times 0.01785}{{0.2}7457861} = {{0.0}10835}}}$${I_{9}(0)} = {\frac{{\lambda_{9}(0)}\beta_{9}{\rho_{9}(0)}}{\sum\limits_{i \in {\{{4,{\text{...,}13},15,16}\}}}{{\lambda_{i}(0)}\beta_{i}{\rho_{i}(0)}}} = {\frac{1 \times 0.5 \times 0.020938}{027457861} = {{0.0}38128}}}$${I_{10}(0)} = {\frac{{\lambda_{10}(0)}\beta_{10}{\rho_{10}(0)}}{\sum\limits_{i \in {\{{4,{\text{...,}13},15,16}\}}}{{\lambda_{i}(0)}\beta_{i}{\rho_{i}(0)}}} = {\frac{1 \times 0.5 \times 0.021232}{0.27457861} = {{0.0}38663}}}$${I_{11}(0)} = {\frac{{\lambda_{11}(0)}\beta_{11}{\rho_{11}(0)}}{\sum\limits_{i \in {\{{4,{\text{...,1}3},15,16}\}}}{{\lambda_{i}(0)}\beta_{i}{\rho_{i}(0)}}} = {\frac{1 \times 1 \times 0.027049}{{0.2}7457861} = {{0.0}98511}}}$${I_{12}(0)} = {\frac{{\lambda_{12}(0)}\beta_{12}{\rho_{12}(0)}}{\sum\limits_{i \in {\{{4,{\text{...,}13},15,16}\}}}{{\lambda_{i}(0)}\beta_{i}{\rho_{i}(0)}}} = {\frac{1 \times 1 \times 0.063}{{0.2}7457861} = {{0.2}29442}}}$${I_{13}(0)} = {\frac{{\lambda_{13}(0)}\beta_{13}{\rho_{13}(0)}}{\sum\limits_{i \in {\{{4,{\text{...,}13},15,16}\}}}{{\lambda_{i}(0)}\beta_{i}{\rho_{i}(0)}}} = {\frac{1 \times 1 \times 0.00792}{{0.2}7457861} = {{0.0}28844}}}$${I_{15}(0)} = {\frac{{\lambda_{15}(0)}\beta_{15}{\rho_{15}(0)}}{\sum\limits_{i \in {\{{4,\text{...},13,15,16}\}}}{{\lambda_{i}(0)}\beta_{i}{\rho_{i}(0)}}} = {\frac{1 \times 1 \times 0.063}{{0.2}7457861} = {{0.2}29442}}}$${I_{16}(0)} = {\frac{{\lambda_{16}(0)}\beta_{16}{\rho_{16}(0)}}{\sum\limits_{i \in {\{{4,{\text{...,}13},15,16}\}}}{{\lambda_{i}(0)}\beta_{i}{\rho_{i}(0)}}} = {\frac{1 \times 1 \times 0.063}{{0.2}7457861} = {{0.2}29442}}}$According to 9, the ranking results of X_(i) are:

index i I_(i)(0) 1 12 0.229442 2 15 0.229442 3 16 0.229442 4 11 0.0985115 7 0.076255 6 10 0.038663 7 9 0.038128 8 13 0.028844 9 5 0.018411 10 80.010835 11 6 0.001482 12 4 0.000544

Take the first 5 X-type variables into detection. Assuming the detectionresults are X_(7,1), X_(11,0), X_(12,1), X_(15,0) and X_(16,0), FIG. 4changes as FIG. 8, in which, X₁₅ and X₁₆ are cause-specific variablesfor BX_(3,1) and all the detection results are negative (variables areall at state 0), thus we know that the state of BX₃ is BX_(3,0).

According to the parameters in Example 1, FIG. 8 is simplified as FIG. 9by using the aforementioned simplification rules 2, 3 and 5, andE(1)=E⁺(0)E(0)=E⁺(0)=X_(7,1)X_(11,0)X_(12,1). Based on E(1) and similarto example 1, FIG. 9 can be divided and simplified as FIG. 10 and FIG.11 according to the DUCG algorithms in [8].

According to the algorithms in [4]-[9], calculate the probabilitiesζ_(i)(y)=ζ_(i)(1) of the sub-graph FIG. 9 and FIG. 10:

Based on FIG. 10,

$\begin{matrix}{{\zeta_{1}(1)} = {\Pr\left\{ {E(1)} \right\}}} \\{= {\Pr\left\{ {X_{7,1}X_{{11},0}X_{12,1}} \right\}}} \\{= {\Pr\left\{ {X_{7,1}X_{12,1}} \right\}}} \\{= {\Pr\left\{ {\frac{r_{7;4}}{r_{7}}A_{7,{1;4}}\frac{r_{4;1}}{r_{4}}A_{4;1}BX_{1}X_{12,1}} \right\}}} \\{= {\Pr\left\{ {\frac{1}{1}A_{7,{1;4}}\frac{1}{1}{A_{4;1}\left( {{\frac{r_{1;1}}{r_{1}}A_{1;1}B_{1}} + {\frac{r_{1;12}}{r_{1}}A_{{1;12},1}X_{12,1}}} \right)}X_{12,1}} \right\}}} \\{= {\Pr\left\{ {A_{7,{1;4}}{A_{4;1}\left( {{\frac{1}{2}A_{1;1}B_{1}} + {\frac{1}{2}A_{{1;12},1}}} \right)}X_{12,1}} \right\}}} \\{= {\Pr\left\{ {A_{7,{1;4}}{A_{4;1}\left( {{\frac{1}{2}A_{1;1}B_{1}} + {\frac{1}{2}A_{{1;12},1}}} \right)}A_{{12},{1;D}}} \right\}}} \\{= {a_{7,{1;4}}{a_{4;1}\left( {{\frac{1}{2}a_{1;1}b_{1}} + {\frac{1}{2}a_{{1;12},1}}} \right)}a_{12,{1;D}}}} \\{= {\left( {- 0.7} \right)\begin{pmatrix} - & - \\ - & 0.9\end{pmatrix}\left( {{\frac{1}{2}\ \begin{pmatrix} - & - \\ - & 1\end{pmatrix}\begin{pmatrix} - \\0.08\end{pmatrix}} + {\frac{1}{2}\begin{pmatrix} - \\0.7\end{pmatrix}}} \right)0.1}} \\{= {0.02457}}\end{matrix}$Based on FIG. 11,

$\begin{matrix}{{\zeta_{2}(1)} = {\Pr\left\{ {E(1)} \right\}}} \\{= {\Pr\left\{ {X_{7,1}X_{11,0}X_{12,1}} \right\}}} \\{= {\Pr\left\{ {\left( {\frac{r_{7;4}}{r_{7}}A_{7,{1;4}}\frac{r_{4;2}}{r_{4}}A_{4;2}{BX}_{2}} \right)\left( {\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{6;2}{BX}_{2}} \right)A_{12,{1;D}}} \right\}}} \\{= {\Pr\left\{ {\left( {\frac{1}{1}A_{7,{1;4}}\frac{1}{1}A_{4;2}} \right)*\left( {\frac{1}{1}A_{11,{0;6}}\frac{1}{1}A_{6;2}} \right)A_{2;2}B_{2}A_{12,{1;D}}} \right\}}} \\{= {\left( {a_{7,{1;4}}a_{4;2}} \right)*\left( {a_{11,{0;6}}a_{6;2}} \right)a_{2;2}b_{2}a_{12,{1;D}}}} \\{= {\left( {\begin{pmatrix} - & 0.7\end{pmatrix}\begin{pmatrix} - & - & - \\ - & 0.5 & 0.5\end{pmatrix}} \right)*\left( \begin{pmatrix}1 & {1 - 0.3} & {1 - 0.8}\end{pmatrix} \right.}} \\{\left. \begin{pmatrix} - & - & - \\ - & 0.6 & 0.3 \\ - & 0.4 & 0.7\end{pmatrix} \right)\begin{pmatrix} - & - & - \\ - & 1 & - \\ - & - & 1\end{pmatrix}\begin{pmatrix} - \\0.01 \\0.02\end{pmatrix}0.1} \\{= 0.00042}\end{matrix}$In which, the original parameter a_(11,0;6)=( - - - ) is modified asa_(11,0;6)=(1 1-0.3 1-0.8), because X_(11,0) is a negative evidence,which means none of the abnormal states occurs and X_(11,0)=1−X_(11,1)according to [4].

According to the algorithms in [8], the weighting coefficients

${\xi_{i}(y)} = \frac{\zeta_{i}(y)}{\sum\limits_{j}{\zeta_{j}(y)}}$of the sub-graphs are:

${\xi_{1}(1)} = {\frac{\zeta_{1}(1)}{{\zeta_{1}(1)} + {\zeta_{2}(1)}} = {\frac{{0.0}2457}{{{0.0}2457} + {{0.0}0042}} = {{0.9}832}}}$${\xi_{2}(1)} = {\frac{\zeta_{2}(1)}{{\zeta_{1}(1)} + {\zeta_{2}(1)}} = {\frac{{0.0}0042}{{{0.0}2457} + {0.00042}} = {{0.0}1681}}}$According to 1 and based on the DUCG algorithms in [4]-[9], wheny=0+1=1, the state probability h_(kj) ^(s)(y)=h_(kj) ^(s)(1) of H_(kj)is:

$\begin{matrix}{{h_{1,1}^{s}(1)} = {{\xi_{1}(1)}\Pr\left\{ {{BX}_{1,1}❘{E(1)}} \right\}}} \\{= {{\xi_{1}(1)}\Pr\left\{ {{BX}_{1,1}❘{X_{7,1}X_{11,0}X_{12,1}}} \right\}}} \\{= {{\xi_{1}(1)}\frac{\Pr\left\{ {{BX}_{1,1}X_{7,1}X_{11,0}X_{12,1}} \right\}}{\Pr\left\{ {X_{7,1}X_{11,0}X_{12,1}} \right\}}}} \\{= {0.983193 \times \frac{0.00022932}{0.0003574}}} \\{= {0.983193 \times 0.64163}} \\{= 0.63085}\end{matrix}$In which

${\Pr\left\{ {X_{7,1}X_{11,0}X_{12,1}} \right\}} = {{\Pr\left\{ {\frac{r_{7;4}}{r_{7}}{A_{7,{1;4}}\left( {{\frac{r_{4;1}}{r_{4}}A_{4;1}{BX}_{1}} + {\frac{r_{4;2}}{r_{4}}A_{4;2}{BX}_{2}}} \right)}\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{6;2}{BX}_{2}X_{12,1}} \right\}} = {{\Pr\begin{Bmatrix}{\frac{r_{7;4}}{r_{7}}A_{7,{1;4}}\frac{r_{4;1}}{r_{4}}A_{4;1}{BX}_{1}\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{6;2}{BX}_{2}X_{12,1}} \\{{+ {{BX}_{1,1}\left( {\frac{r_{7;4}}{r_{7}}A_{7,{1;4}}\frac{r_{4;2}}{r_{4}}A_{4;2}} \right)}}*\left( {\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{6;2}} \right){BX}_{2}X_{12,1}}\end{Bmatrix}} = {{\Pr\left\{ {\begin{pmatrix}{\frac{r_{7;4}}{r_{7}}A_{7,{1;4}}\frac{r_{4;1}}{r_{4}}{A_{4;1}\begin{pmatrix}{\frac{r_{1;1}}{r_{1}}A_{1;1}B_{1}} \\{{+ \frac{r_{1;12}}{r_{1}}}A_{1;12}}\end{pmatrix}}\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{6;2}} \\{{+ \left( {\frac{r_{7;4}}{r_{7}}A_{7,{1;4}}\frac{r_{4;2}}{r_{4}}A_{4;2}} \right)}*\left( {\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{6;2}} \right)}\end{pmatrix}\frac{r_{2;2}}{r_{2}}A_{2;2}B_{2}X_{12,1}} \right\}} = {{\begin{pmatrix}{\left( {\frac{r_{7;4}}{r_{7}}a_{7,{1;4}}\frac{r_{4;1}}{r_{4}}{a_{4;1}\begin{pmatrix}{\frac{r_{1;1}}{r_{1}}a_{1;1}b_{1,1}} \\{{+ \frac{r_{1;12}}{r_{1}}}a_{1,{;12}}}\end{pmatrix}}} \right)\left( {\frac{r_{11;6}}{r_{11}}a_{11,{0;6}}\frac{r_{6;2}}{r_{6}}a_{6;2}} \right)} \\{{+ \left( {\frac{r_{7;4}}{r_{7}}a_{7,{1;4}}\frac{r_{4;2}}{r_{4}}A_{4;2}} \right)}*\left( {\frac{r_{11;6}}{r_{11}}a_{11,{0;6}}\frac{r_{6;2}}{r_{6}}a_{6;2}} \right)}\end{pmatrix}a_{2;2}b_{2}a_{12,{1;D}}} = {{\begin{pmatrix}{\left( {\begin{pmatrix} - & 0.7\end{pmatrix}\frac{1}{2}\begin{pmatrix} - & - \\ - & 0.9\end{pmatrix}\left( {{\frac{1}{2}\begin{pmatrix} - & - \\ - & 1\end{pmatrix}\begin{pmatrix} - \\0.08\end{pmatrix}} + {\frac{1}{2}\begin{pmatrix} - \\0.08\end{pmatrix}}} \right)} \right) \cdot \left( {\begin{pmatrix}1 & {1 - 0.3} & {1 - 0.8}\end{pmatrix}\begin{pmatrix} - & - & - \\ - & 0.6 & 0.3 \\ - & 0.4 & 0.7\end{pmatrix}} \right)} \\{{+ \left( {\begin{pmatrix} - & 0.7\end{pmatrix}\frac{1}{2}\begin{pmatrix} - & - & - \\ - & 0.5 & 0.5\end{pmatrix}} \right)}*\left( {\begin{pmatrix}1 & {1 - 0.3} & {1 - 0.8}\end{pmatrix}\begin{pmatrix} - & - & - \\ - & 0.6 & 0.3 \\ - & 0.4 & 0.7\end{pmatrix}} \right)}\end{pmatrix}\begin{pmatrix} - & - & - \\ - & 1 & - \\ - & - & 1\end{pmatrix}\begin{pmatrix} - \\0.01 \\0.02\end{pmatrix}0.1}\; = {{\left( {{0.12285 \times \begin{pmatrix} - & 0.5 & 0.35\end{pmatrix}} + {0.39\begin{pmatrix} - & 0.175 & 0.175\end{pmatrix}*\begin{pmatrix} - & 0.5 & 0.35\end{pmatrix}}} \right)\begin{pmatrix} - \\0.01 \\0.02\end{pmatrix}0.1}\; = {{\begin{pmatrix} - & 0.148925 & 0.104248\end{pmatrix}\begin{pmatrix} - \\0.01 \\0.02\end{pmatrix}0.1}\; = \; 0.0003574}}}}}}}$

${\Pr\left\{ {{BX}_{1,1}X_{7,1}X_{11,0}X_{12,1}} \right\}} = {{\Pr\left\{ {{BX}_{1,1}\frac{r_{7;4}}{r_{7}}{A_{7,{1;4}}\left( {{\frac{r_{4;1}}{r_{4}}A_{4;1}{BX}_{1}} + {\frac{r_{4;2}}{r_{4}}A_{4;2}{BX}_{2}}} \right)}\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{6;2}{BX}_{2}A_{12,{1;D}}} \right\}}\; = {{\Pr\begin{Bmatrix}{\frac{r_{7;4}}{r_{7}}A_{7,{1;4}}\frac{r_{4;1}}{r_{4}}A_{{4;1},1}{BX}_{1,1}\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{6;2}{BX}_{2}A_{12,{1;D}}} \\{{+ {{BX}_{1,1}\left( {\frac{r_{7;4}}{r_{7}}A_{7,{1;4}}\frac{r_{4;2}}{r_{4}}A_{4;2}} \right)}}*\left( {\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{6;2}} \right){BX}_{2}A_{12,{1;D}}}\end{Bmatrix}}\; = {{\Pr\left\{ {\begin{pmatrix}{\frac{r_{7;4}}{r_{7}}A_{7,{1;4}}\frac{r_{4;1}}{r_{4}}{A_{{4;1},1}\begin{pmatrix}{\frac{r_{1;1}}{r_{1}}A_{1,{1;1},1}B_{1,1}} \\{{+ \frac{r_{1;12}}{r_{1}}}A_{1,{1;12}}}\end{pmatrix}}\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{6;2}} \\{{+ \begin{pmatrix}{\frac{r_{1;1}}{r_{1}}A_{1,{1;1},1}B_{1,1}} \\{{+ \frac{r_{1;12}}{r_{1}}}A_{1,{1;12}}}\end{pmatrix}}\left( {\frac{r_{7;4}}{r_{7}}A_{7,{1;4}}\frac{r_{4;2}}{r_{4}}A_{4;2}} \right)*\left( {\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{6;2}} \right)}\end{pmatrix}\frac{r_{2;2}}{r_{2}}A_{2;2}B_{2}A_{12,{1;D}}} \right\}}\; = {{\begin{pmatrix}{\left( {\frac{r_{7;4}}{r_{7}}a_{7,{1;4}}\frac{r_{4;1}}{r_{4}}{a_{{4;1},1}\begin{pmatrix}{\frac{r_{1;1}}{r_{1}}a_{1,{1;1},1}b_{1,1}} \\{{+ \frac{r_{1;12}}{r_{1}}}a_{1,{1;12},1}}\end{pmatrix}}} \right)\left( {\frac{r_{11;6}}{r_{11}}a_{11,{0;6}}\frac{r_{6;2}}{r_{6}}a_{6;2}} \right)} \\{{+ \begin{pmatrix}{\frac{r_{1;1}}{r_{1}}a_{1,{1;1},1}b_{1,1}} \\{{+ \frac{r_{1;12}}{r_{1}}}a_{1,{1;12},1}}\end{pmatrix}}\left( {\frac{r_{7;4}}{r_{7}}a_{7,{1;4}}\frac{r_{4;2}}{r_{4}}a_{4;2}} \right)*\left( {\frac{r_{11;6}}{r_{11}}a_{11,{0;6}}\frac{r_{6;2}}{r_{6}}a_{6;2}} \right)}\end{pmatrix}a_{2;2}b_{2}a_{12,{1;D}}}\; = {{\begin{pmatrix}{\left( {\begin{pmatrix} - & 0.7\end{pmatrix}\frac{1}{2}\begin{pmatrix} - & - \\ - & 0.9\end{pmatrix}\left( {{\frac{1}{2} \times 0.08} + {\frac{1}{2} \times 0.7}} \right)} \right)\left( {\begin{pmatrix}1 & {1 - 0.3} & {1 - 0.8}\end{pmatrix}\begin{pmatrix} - & - & - \\ - & 0.6 & 0.3 \\ - & 0.4 & 0.7\end{pmatrix}} \right)} \\{{+ \left( {{\frac{1}{2} \times 0.08} + {\frac{1}{2} \times 0.7}} \right)}\left( {\begin{pmatrix} - & 0.7\end{pmatrix}\frac{1}{2}\begin{pmatrix} - & - & - \\ - & 0.5 & 0.5\end{pmatrix}} \right)*\left( {\begin{pmatrix}1 & {1 - 0.3} & {1 - 0.8}\end{pmatrix}\begin{pmatrix} - & - & - \\ - & 0.6 & 0.3 \\ - & 0.4 & 0.7\end{pmatrix}} \right)}\end{pmatrix}\begin{pmatrix} - \\0.01 \\0.02\end{pmatrix}0.1}\; = {{\left( {{0.12285\begin{pmatrix} - & 0.5 & 0.35\end{pmatrix}} + {0.39\begin{pmatrix} - & 0.175 & 0.175\end{pmatrix}*\begin{pmatrix} - & 0.5 & 0.35\end{pmatrix}}} \right)\begin{pmatrix} - \\0.01 \\0.02\end{pmatrix}0.1}\; = \;{{\begin{pmatrix} - & 0.09555 & 0.066885\end{pmatrix}\begin{pmatrix} - \\0.01 \\0.02\end{pmatrix}0.1}\; = \; 0.00022932}}}}}}}$${{{In}\mspace{14mu}{the}\mspace{14mu}{same}\mspace{14mu}{way}},\begin{matrix}{{h_{2,1}^{s}(1)} = {{\xi_{2}(1)}\Pr\left\{ {{BX}_{2,1}❘{E(1)}} \right\}}} \\{= {{\xi_{2}(1)}\Pr\left\{ {{BX}_{2,1}❘{X_{7,1}X_{11,0}X_{12,1}}} \right\}}} \\{= 0.007}\end{matrix}}\mspace{14mu}$

$\begin{matrix}{{h_{2,2}^{s}(1)} = {{\xi_{2}(1)}\Pr\left\{ {{BX}_{2,2}❘{E(1)}} \right\}}} \\{= {{\xi_{2}(1)}\Pr\left\{ {{BX}_{2,2}❘{X_{7,1}X_{11,0}X_{12,1}}} \right\}}} \\{= 0.009781}\end{matrix}$

Based on

${{h_{kj}^{r}(1)} = \frac{h_{kj}^{s}(1)}{\sum\limits_{H_{kj} \in {S_{H}{(1)}}}{h_{kj}^{s}(1)}}},$the rank probabilities are:

index H_(kj) h_(kj) ^(r) (1) 1 BX_(1, 1) 0.974089 2 BX_(2, 2) 0.015103 3BX_(2, 1) 0.010809Compared to the rank probabilities h_(kj) ^(r)(0) before detections, wesee (1) BX_(3,1) which ranks first is eliminated and the possible resultis a reduced state space S_(H)(1), and (2) the rank probability ofBX_(1,1) is far bigger than the rest two, which means that the cause ofabnormality in the real target system can be verified as BX_(1,1) withonly five X-type variables being detected.

Example 2: y=1

According to 2, when no evidence (y=0), based on FIG. 4,S_(H)(0)={H_(1,1), H_(2,1), H_(2,2), H_(3,1)}={BX_(1,1), BX_(2,1),BX_(2,2), BX_(3,1)}, and the state pending X-type variables are {X₄, X₅,X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₅, X₁₆}. All these states areunknown and single-connect to H_(k) in S_(H)(0) without state-knownblocking variables, so that S_(X)(0)={4, 5, 6, 7, 8, 9, 10, 11, 12, 13,15, 16}. Since X₁₅ and X₁₆ are cause-specific variable for BX_(3,1),they are eliminated from S_(X)(0), that means S_(Xs)(0)={4, 5, 6, 7, 8,9, 10, 11, 12, 13}. Accordingly,S_(4G)(0)=S_(5G)(0)=S_(7G)(0)=S_(8G)(0)=S_(9G)(0)=S_(10G)(0)=S_(11G)(0)=S_(12G)(0)=S_(13G)(0)={1}and S_(6G)(0)={1, 2}.

As example 1 and based on FIG. 9, E(1)=X_(7,1)X_(11,0)X_(12,1),S_(H)(1)={H_(1,1), H_(2,1), H_(2,2)}={BX_(1,1), BX_(2,1), BX_(2,2)} andthe testing X variables are {X₄, X₅, X₆, X₈, X₉, X₁₀}. All of them arestates unknown and single-connect to like H_(k)∈{BX₁, BX₂} withoutstate-known blocking variables. According to 9, X₄ is the only upstreamvariable of X_(7,1) and can be eliminated from the ranking, thusS_(X)(1)={5, 6, 8, 9, 10}. Since there is no cause-specific variable,then S_(Xs)(1)=S_(X)(1). As the same as y=0, one can getS_(4G)(1)=S_(5G)(1)=S_(8G)(1)=S_(9G)(1)=S_(10G)(1)={1} and S_(6G)(1)={1,2}.

According to 3 and based on FIG. 9, one can get the following equations:S_(13K)(1)={1},S_(4K)(1)=S_(5K)(1)=S_(8K)(1)=S_(9K)(1)=S_(10K)(1)={1,2}, andS_(6K)(1)={2}; and accordingly m₆(1)=1, m₄(1)=m₅(1)=m₈(1)=m₁₀(1)m₉(1)=2, S_(1J)(1)={1}, and S_(2J)(1)={2}. Still same as y=0, ω_(kj)=1(k∈{1,2}).

According to 5, take the following formula for calculation as same as inexample 1:

${\rho_{i}(y)} = {\frac{1}{m_{i}(y)}{\sum\limits_{k \in {S_{iK}{(y)}}}{\sum\limits_{j \in {S_{kJ}{(y)}}}{\omega_{kj}{\sum\limits_{g \in {S_{iG}{(y)}}}{\Pr\left\{ X_{ig} \middle| {E(y)} \right\}{{{\Pr\left\{ H_{kj} \middle| {X_{ig}{E(y)}} \right\}} - {\Pr\left\{ H_{kj} \middle| {E(y)} \right\}}}}}}}}}}$where i∈S_(Xs)(0)={4, 5, 6, 8, 9, 10}. Note m′_(i)(y)=m_(i)(y) in thisexample.

According to FIG. 9, since ω_(kj)=1 and k∈{1, 2}, we have

$\begin{matrix}{{\rho_{4}(1)} = {\frac{1}{m_{4}(1)}{\sum\limits_{k \in {\{{1,2}\}}}{\sum\limits_{j \in {S_{kJ}{(1)}}}{\sum\limits_{g \in {\{ 1\}}}{\Pr\left\{ X_{ig} \middle| {E(1)} \right\}{{{\Pr\left\{ H_{kj} \middle| {X_{ig}{E(1)}} \right\}} - {\Pr\left\{ H_{kj} \middle| {E(1)} \right\}}}}}}}}}} \\{= {\frac{1}{2}\Pr\left\{ X_{4,1} \middle| {E(1)} \right\}\begin{pmatrix}{{{\Pr\left\{ H_{1,1} \middle| {X_{4,1}{E(1)}} \right\}} - {\Pr\left\{ H_{1,1} \middle| {E(1)} \right\}}}} \\{+ {{{\Pr\left\{ H_{2,1} \middle| {X_{4,1}{E(1)}} \right\}} - {\Pr\left\{ H_{2,1} \middle| {E(1)} \right\}}}}} \\{+ {{{\Pr\left\{ H_{2,2} \middle| {X_{4,1}{E(1)}} \right\}} - {\Pr\left\{ H_{2,2} \middle| {E(1)} \right\}}}}}\end{pmatrix}}} \\{= {\frac{1}{2}\Pr\left\{ X_{4,1} \middle| {E(1)} \right\}\begin{pmatrix}{{{\Pr\left\{ {BX}_{1,1} \middle| {X_{4,1}{E(1)}} \right\}} - {\Pr\left\{ {BX}_{1,1} \middle| {E(1)} \right\}}}} \\{+ {{{\Pr\left\{ {BX}_{2,1} \middle| {X_{4,1}{E(1)}} \right\}} - {\Pr\left\{ {BX}_{2,1} \middle| {E(1)} \right\}}}}} \\{+ {{{\Pr\left\{ {BX}_{2,2} \middle| {X_{4,1}{E(1)}} \right\}} - {\Pr\left\{ {BX}_{2,2} \middle| {E(1)} \right\}}}}}\end{pmatrix}}} \\{= {\frac{1}{2}\frac{1}{\Pr\left\{ {E(1)} \right\}}\begin{pmatrix}{{{\Pr\left\{ {{BX}_{1,1}X_{4,1}{E(1)}} \right\}} - {\Pr\left\{ {X_{4,1}{E(1)}} \right\}\Pr\left\{ B_{1,1} \middle| {E(1)} \right\}}}} \\{+ {{{\Pr\left\{ {{BX}_{2,1}X_{4,1}{E(1)}} \right\}} - {\Pr\left\{ {X_{4,1}{E(1)}} \right\}\Pr\left\{ B_{2,1} \middle| {E(1)} \right\}}}}} \\{+ {{{\Pr\left\{ {{BX}_{2,2}X_{4,1}{E(1)}} \right\}} - {\Pr\left\{ {X_{4,1}{E(1)}} \right\}\Pr\left\{ B_{2,2} \middle| {E(1)} \right\}}}}}\end{pmatrix}}}\end{matrix}$

In which

$\begin{matrix}{{\Pr\left\{ {X_{4,1}{E(1)}} \right\}} = {\Pr\left\{ {X_{4,1}X_{7,1}X_{{11},0}X_{{12},1}} \right\}}} \\{= {{\Pr\left\{ {\frac{r_{7;4}}{r_{7}}A_{7,{1;4}}X_{4}X_{4,1}X_{{11},0}X_{{12},1}} \right\}} = {\Pr\left\{ {\frac{r_{7;4}}{r_{7}}A_{7,{1;4}}X_{4,1}X_{{11},0}X_{12,1}} \right\}}}} \\{= {\Pr\left\{ {\frac{r_{7;4}}{r_{7}}{A_{7,{1;4},1}\left( {{\frac{r_{4;1}}{r_{4}}A_{4,{1;1}}{BX}_{1}} + {\frac{r_{4;2}}{r_{4}}A_{4,{1;2}}BX_{2}}} \right)}\frac{r_{11;6}}{r_{11}}A_{{11},{0;6}}\frac{r_{6;2}}{r_{ó}}A_{6;2}{BX}_{2}A_{12,{1;D}}} \right\}}} \\{= {\Pr\begin{Bmatrix}{\frac{r_{7;4}}{r_{7}}A_{7,{1;4},1}\frac{r_{4;1}}{r_{4}}A_{4,{1;1}}BX_{1}\frac{r_{11;ó}}{r_{11}}A_{{11},{0;6}}\frac{r_{6;2}}{r_{6}}A_{6;2}BX_{2}A_{12,{1;D}}} \\{{+ \left( {\frac{r_{7;4}}{r_{7}}A_{7,{1;4},1}\frac{r_{4;2}}{r_{4}}A_{4,{1;2}}} \right)}*\left( {\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{6;2}} \right)BX_{2}A_{12,{1;D}}}\end{Bmatrix}}} \\{= {\Pr\left\{ {\begin{pmatrix}{\frac{r_{7;4}}{r_{7}}A_{7,{1;4},1}\frac{r_{4;1}}{r_{4}}{A_{4,{1;1}}\begin{pmatrix}{\frac{r_{1;1}}{r_{1}}A_{1;1}B_{1}} \\{{+ \frac{r_{1;12}}{r_{1}}}A_{1;12}}\end{pmatrix}}\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{6;2}} \\{{+ \left( {\frac{r_{7;4}}{r_{7}}A_{7,{1;4},1}\frac{r_{4;2}}{r_{4}}A_{4,{1;2}}} \right)}*\left( {\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{6;2}} \right)}\end{pmatrix}\frac{r_{2;2}}{r_{2}} A_{2;2} B_{2} A_{12,{1;D}}} \right\}}} \\{= {\begin{pmatrix}{\left( {\frac{r_{7;4}}{r_{7}}a_{7,{1;4},1}\frac{r_{4;1}}{r_{4}}{a_{4,{1;1}}\begin{pmatrix}{\frac{r_{1;1}}{r_{1}}a_{1;1}b_{1}} \\{{+ \frac{r_{1;12}}{r_{1}}}a_{1;12}}\end{pmatrix}}} \right)\left( {\frac{r_{11;6}}{r_{11}}a_{11,{0;6}}\frac{r_{6;2}}{r_{6}}a_{6;2}} \right)} \\{{+ \left( {\frac{r_{7;4}}{r_{7}}a_{7,{1;4},1}\frac{r_{4;2}}{r_{4}}A_{4,{1;2}}} \right)}*\left( {\frac{r_{11;6}}{r_{11}}a_{11,{0;6}}\frac{r_{6;2}}{r_{6}}a_{6;2}} \right)}\end{pmatrix}a_{2;2}b_{2}a_{12,{1;D}}}} \\{= {\begin{pmatrix}{\left( {0.7\frac{1}{2}\begin{pmatrix} - & 0.9\end{pmatrix}\begin{pmatrix} - \\0.39\end{pmatrix}} \right)\left( {\begin{pmatrix}1 & {1 - 0.3} & {1 - 0.8}\end{pmatrix}\begin{pmatrix} - & - & - \\ - & 0.6 & 0.3 \\ - & 0.4 & 0.7\end{pmatrix}} \right)} \\{{+ \left( {0.7\frac{1}{2}\begin{pmatrix} - & 0.5 & 0.5\end{pmatrix}} \right)}*\left( {\begin{pmatrix}1 & {1 - 0.3} & {1 - 0.8}\end{pmatrix}\begin{pmatrix} - & - & - \\ - & 0.6 & 0.3 \\ - & 0.4 & 0.7\end{pmatrix}} \right)}\end{pmatrix}\begin{pmatrix} - \\0.01 \\0.02\end{pmatrix}0.1}} \\{= 0.00035742}\end{matrix}$

$\begin{matrix}{{\Pr\left\{ {{BX}_{1,1}X_{4,1}{E(1)}} \right\}} = {\Pr\left\{ {{BX}_{1,1}X_{4,1}X_{7,1}X_{{11},0}X_{{12},1}} \right\}}} \\{= {{\Pr\left\{ {\frac{r_{7;4}}{r_{7}}A_{7,{1;4}}X_{4}X_{4,1}X_{{11},0}X_{{12},1}{BX}_{1,1}} \right\}} = {\Pr\left\{ {\frac{r_{7;4}}{r_{7}}A_{7,{1;4}}X_{4,1}X_{{11},0}X_{12,1}{BX}_{1,1}} \right\}}}} \\{= {\Pr\left\{ {\frac{r_{7;4}}{r_{7}}{A_{7,{1;4},1}\left( {{\frac{r_{4;1}}{r_{4}}A_{4,{1;1},1}{BX}_{1,1}} + {\frac{r_{4;2}}{r_{4}}A_{4,{1;2}}BX_{2}{BX}_{1,1}}} \right)}\frac{r_{11;6}}{r_{11}}A_{{11},{0;6}}\frac{r_{6;2}}{r_{6}}A_{6;2}{BX}_{2}A_{12,{1;D}}} \right\}}} \\{= {\Pr\begin{Bmatrix}{\frac{r_{7;4}}{r_{7}}A_{7,{1;4},1}\frac{r_{4;1}}{r_{4}}A_{4,{1;1},1}BX_{1,1}\frac{r_{11;6}}{r_{11}}A_{{11},{0;6}}\frac{r_{6;2}}{r_{6}}A_{6;2}BX_{2}A_{12,{1;D}}} \\{{+ \left( {\frac{r_{7;4}}{r_{7}}A_{7,{1;4},1}\frac{r_{4;2}}{r_{4}}A_{4,{1;2}}} \right)}*\left( {\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{ó}}A_{6;2}} \right)BX_{2}{BX}_{1,1}A_{12,{1;D}}}\end{Bmatrix}}} \\{= {\Pr\left\{ {\begin{pmatrix}{\frac{r_{7;4}}{r_{7}}A_{7,{1;4},1}\frac{r_{4;1}}{r_{4}}{A_{4,{1;1},1}\begin{pmatrix}{\frac{r_{1;1}}{r_{1}}A_{1,{1;1}}B_{1}} \\{{+ \frac{r_{1;12}}{r_{1}}}A_{1,{1;12}}}\end{pmatrix}}\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{6;2}} \\{{+ \left( {\frac{r_{7;4}}{r_{7}}A_{7,{1;4},1}\frac{r_{4;2}}{r_{4}}A_{4,{1;2}}} \right)}*\left( {\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{6;2}} \right)\begin{pmatrix}{{\frac{r_{1;1}}{r_{1}}A_{1,{1;1}}B_{1}} +} \\{\frac{r_{1;12}}{r_{1}}A_{1,{1;12}}}\end{pmatrix}}\end{pmatrix}\frac{r_{2;2}}{r_{2}} A_{2;2} B_{2} A_{12,{1;D}}} \right\}}} \\{= {\begin{pmatrix}{\left( {\frac{r_{7;4}}{r_{4}}a_{7,{1;4},1}\frac{r_{4;1}}{r_{4}}{a_{4,{1;1},1}\begin{pmatrix}{\frac{r_{1;1}}{r_{1}}a_{1,{1;1}}b_{1}} \\{{+ \frac{r_{1;12}}{r_{1}}}a_{1,{1;12}}}\end{pmatrix}}} \right)\left( {\frac{r_{11;6}}{r_{11}}a_{11,{0;6}}\frac{r_{6;2}}{r_{6}}a_{6;2}} \right)} \\{{+ \left( {\frac{r_{7;4}}{r_{7}}a_{7,{1;4},1}\frac{r_{4;2}}{r_{4}}A_{4,{1;2}}} \right)}*\left( {\frac{r_{11;6}}{r_{11}}a_{11,{0;6}}\frac{r_{6;2}}{r_{6}}a_{6;2}} \right)\begin{pmatrix}{{\frac{r_{1;1}}{r_{1}}a_{1,{1;1}}b_{1}} +} \\{\frac{r_{1;12}}{r_{1}}a_{1,{1;12}}}\end{pmatrix}}\end{pmatrix}a_{2;2}b_{2}a_{12,{1;D}}}} \\{= {\begin{pmatrix}{\left( {0.7\frac{1}{2}0.9(0.39)} \right)\left( {\begin{pmatrix}1 & {1 - 0.3} & {1 - 0.8}\end{pmatrix}\begin{pmatrix} - & - & - \\ - & 0.6 & 0.3 \\ - & 0.4 & 0.7\end{pmatrix}} \right)} \\{{+ \left( {0.7\frac{1}{2}\begin{pmatrix} - & 0.5 & 0.5\end{pmatrix}} \right)}*\left( {\begin{pmatrix}1 & {1 - 0.3} & {1 - 0.8}\end{pmatrix}\begin{pmatrix} - & - & - \\ - & 0.6 & 0.3 \\ - & 0.4 & 0.7\end{pmatrix}} \right)(0.39)}\end{pmatrix}\begin{pmatrix} - \\0.01 \\0.02\end{pmatrix}0.1}} \\{= 0.0002293}\end{matrix}$

$\begin{matrix}{{\Pr\left\{ {{BX}_{2,1}X_{4,1}{E(1)}} \right\}} = {\Pr\left\{ {{BX}_{2,1}X_{4,1}X_{7,1}X_{11,0}X_{12,1}} \right\}}} \\{= {{\Pr\left\{ {\frac{r_{7;4}}{r_{7}}A_{7,{1;4}}X_{4}X_{4,1}X_{11,0}X_{12,1}{BX}_{2,1}} \right\}} = {\Pr\left\{ {\frac{r_{7;4}}{r_{7}}A_{7,{1;4}}X_{4,1}X_{11,0}X_{12,1}{BX}_{2,1}} \right\}}}} \\{= {\Pr\left\{ {\frac{r_{7;4}}{r_{7}}{A_{7,{1;4}}\left( {{\frac{r_{4;1}}{r_{4}}A_{4,{1;1}}{BX}_{1}} + {\frac{r_{4;2}}{r_{4}}A_{4,{1;2}}{BX}_{2}}} \right)}\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{6;2}{BX}_{2}A_{12,{1;D}}{BX}_{2,1}} \right\}}} \\{= {\Pr\begin{Bmatrix}{\frac{r_{7;4}}{r_{7}}A_{7,{1;4},1}\frac{r_{4;1}}{r_{4}}A_{4,{1;1}}{BX}_{1}\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{{6;2},1}{BX}_{2,1}A_{12,{1;D}}} \\{{+ \left( {\frac{r_{7;4}}{r_{7}}A_{7,{1;4},1}\frac{r_{4;2}}{r_{4}}A_{4,{1;2},1}} \right)}\left( {\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{{6;2},1}} \right){BX}_{2,1}A_{112,{1;D}}}\end{Bmatrix}}} \\{= {P r\left\{ {\begin{pmatrix}{\frac{r_{7;4}}{r_{7}}A_{7,{1;4},1}\frac{r_{4;1}}{r_{4}}{A_{4,{1;1}}\begin{pmatrix}{\frac{r_{1;1}}{r_{1}}A_{1;1}B_{1}} \\{{+ \frac{r_{1;12}}{r_{1}}}A_{1;12}}\end{pmatrix}}\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{{6;2},1}} \\{{+ \left( {\frac{r_{7;4}}{r_{7}}A_{7,{1;4},1}\frac{r_{4;2}}{r_{4}}A_{4,{1;2},1}} \right)}\left( {\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{{6;2},1}} \right)}\end{pmatrix}\frac{r_{2;2}}{r_{2}} A_{2,{1;2}} B_{2} A_{12,{1;D}}} \right\}}} \\{= {\begin{pmatrix}{\left( {\frac{r_{7;4}}{r_{7}}a_{7,{1;4},1}\frac{r_{4;1}}{r_{4}}{a_{4,{1;1}}\begin{pmatrix}{\frac{r_{1;1}}{r_{1}}a_{1;1}b_{1}} \\{{+ \frac{r_{1;12}}{r_{1}}}a_{1;12}}\end{pmatrix}}} \right)\left( {\frac{r_{11;6}}{r_{11}}a_{11,{0;6}}\frac{r_{6;2}}{r_{6}}a_{{6;2},1}} \right)} \\{{+ \left( {\frac{r_{7;4}}{r_{7}}a_{7,{1;4},1}\frac{r_{4;2}}{r_{4}}A_{4,{1;2},1}} \right)}\left( {\frac{r_{11;6}}{r_{11}}a_{11,{0;6}}\frac{r_{6;2}}{r_{6}}a_{{6;2},1}} \right)}\end{pmatrix}a_{2,{1;2}}b_{2}a_{12,{1;D}}}} \\{= {\begin{pmatrix}{\left( {0.7\frac{1}{2}\begin{pmatrix} - & 0.9\end{pmatrix}\begin{pmatrix} - \\0.39\end{pmatrix}} \right)\left( {\begin{pmatrix}1 & {1 - 0.3} & {1 - 0.8}\end{pmatrix}\begin{pmatrix} - \\0.6 \\0.4\end{pmatrix}} \right)} \\{{+ \left( {0.7\frac{1}{2}(0.5)} \right)}\left( {\begin{pmatrix}1 & {1 - 0.3} & {1 - 0.8}\end{pmatrix}\begin{pmatrix} - \\0.6 \\0.4\end{pmatrix}} \right)}\end{pmatrix}(0.01)0.1}} \\{= 0.0001489}\end{matrix}$

$\begin{matrix}{{\Pr\left\{ {{BX}_{2,2}X_{4,1}{E(1)}} \right\}} = {\Pr\left\{ {{BX}_{2,2}X_{4,1}X_{7,1}X_{11,0}X_{12,1}} \right\}}} \\{= {{\Pr\left\{ {\frac{r_{7;4}}{r_{7}}A_{7,{1;4}}X_{4}X_{4,1}X_{11,0}X_{12,1}{BX}_{2,1}} \right\}} = {\Pr\left\{ {\frac{r_{7;4}}{r_{7}}A_{7,{1;4}}X_{4,1}X_{11,0}X_{12,1}{BX}_{2,2}} \right\}}}} \\{= {\Pr\left\{ {\frac{r_{7;4}}{r_{7}}{A_{7,{1;4},1}\left( {{\frac{r_{4;1}}{r_{4}}A_{4,{1;1}}{BX}_{1}} + {\frac{r_{4;2}}{r_{4}}A_{4,{1;2}}{BX}_{2}}} \right)}\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{6;2}{BX}_{2}A_{12,{1;D}}{BX}_{2,2}} \right\}}} \\{= {\Pr\begin{Bmatrix}{\frac{r_{7;4}}{r_{7}}A_{7,{1;4},1}\frac{r_{4;1}}{r_{4}}A_{4,{1;1}}{BX}_{1}\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{{6;2},1}{BX}_{2,2}A_{12,{1;D}}} \\{{+ \left( {\frac{r_{7;4}}{r_{7}}A_{7,{1;4},1}\frac{r_{4;2}}{r_{4}}A_{4,{1;2},2}} \right)}*\left( {\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{{6;2},2}} \right){BX}_{2,2}A_{12,{1;D}}}\end{Bmatrix}}} \\{= {P r\left\{ {\begin{pmatrix}{\frac{r_{7;4}}{r_{7}}A_{7,{1;4},1}\frac{r_{4;1}}{r_{4}}{A_{4,{1;1}}\begin{pmatrix}{\frac{r_{1;1}}{r_{1}}A_{1;1}B_{1}} \\{{+ \frac{r_{1;12}}{r_{1}}}A_{{1;12},1}}\end{pmatrix}}\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{{6;2},2}} \\{{+ \left( {\frac{r_{7;4}}{r_{7}}A_{7,{1;4},1}\frac{r_{4;2}}{r_{4}}A_{4,{1;2},2}} \right)}*\left( {\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{{6;2},2}} \right)}\end{pmatrix}\frac{r_{2;2}}{r_{2}} A_{2,{2;2}} B_{2} A_{12,{1;D}}} \right\}}} \\{= {\begin{pmatrix}{\left( {\frac{r_{7;4}}{r_{7}}a_{7,{1;4},1}\frac{r_{4;1}}{r_{4}}{a_{4,{1;1}}\begin{pmatrix}{\frac{r_{1;1}}{r_{1}}a_{1;1}b_{1}} \\{{+ \frac{r_{1;12}}{r_{1}}}a_{{1;12},1}}\end{pmatrix}}} \right)\left( {\frac{r_{11;6}}{r_{11}}a_{11,{0;6}}\frac{r_{6;2}}{r_{6}}a_{{6;2},2}} \right)} \\{{+ \left( {\frac{r_{7;4}}{r_{7}}a_{7,{1;4},1}\frac{r_{4;2}}{r_{4}}A_{4,{1;2},2}} \right)}*\left( {\frac{r_{11;6}}{r_{11}}a_{11,{0;6}}\frac{r_{6;2}}{r_{6}}a_{{6;2},2}} \right)}\end{pmatrix}a_{2,{2;2}}b_{2}a_{12,{1;D}}}} \\{= {\begin{pmatrix}{\left( {0.7\frac{1}{2}\begin{pmatrix} - & 0.9\end{pmatrix}\begin{pmatrix} - \\0.39\end{pmatrix}} \right)\left( {\begin{pmatrix}1 & {1 - 0.3} & {1 - 0.8}\end{pmatrix}\begin{pmatrix} - \\0.3 \\0.7\end{pmatrix}} \right)} \\{{+ \left( {0.7\frac{1}{2}(0.5)} \right)}*\left( {\begin{pmatrix}1 & {1 - 0.3} & {1 - 0.8}\end{pmatrix}\begin{pmatrix} - \\0.3 \\0.7\end{pmatrix}} \right)}\end{pmatrix}(0.02)0.1}} \\{= 0.000208}\end{matrix}$

Thus

$\begin{matrix}{{\rho_{4}(1)} = {\frac{1}{2}\frac{1}{\Pr\left\{ {E(1)} \right\}}\begin{pmatrix}{{{\Pr\left\{ {{BX}_{1,1}X_{4,1}{E(1)}} \right\}} - {\Pr\left\{ {X_{4,1}{E(1)}} \right\}\Pr\left\{ {{BX}_{1,1}❘{E(1)}} \right\}}}} \\{+ {{{\Pr\left\{ {{BX}_{2,1}X_{4,1}{E(1)}} \right\}} - {\Pr\left\{ {X_{4,1}{E(1)}} \right\}\Pr\left\{ {{BX}_{2,1}❘{E(1)}} \right\}}}}} \\{+ {{{\Pr\left\{ {{BX}_{2,2}X_{4,1}{E(1)}} \right\}} - {\Pr\left\{ {X_{4,1}{E(1)}} \right\}\Pr\left\{ {{BX}_{2,2}❘{E(1)}} \right\}}}}}\end{pmatrix}}} \\{= {\frac{1}{2}\frac{1}{0.0003574}\begin{pmatrix}{{0.0002293 - {0.00035742 \times 0.64163}}} \\{+ {{0.0001489 - {0.00035742 \times 0.41662}}}} \\{+ {{0.000208 - {0.00035742 \times 0.58286}}}}\end{pmatrix}}} \\{= 0}\end{matrix}$

Similarly,

$\begin{matrix}{{\rho_{5}(1)} = {\frac{1}{m_{5}(1)}{\sum\limits_{k \in {{\{{1,2}\}}j}}{\sum\limits_{\in {S_{kJ}{(1)}}}{\sum\limits_{g \in {\{ 1\}}}{Pr\left\{ X_{ig} \middle| {E(1)} \right\}{{{\Pr\left\{ H_{kj} \middle| {X_{ig}{E(1)}} \right\}} - {Pr\left\{ H_{kj} \middle| {E(1)} \right\}}}}}}}}}} \\{= {\frac{1}{2}Pr\left\{ X_{5,1} \middle| {E(1)} \right\}\begin{pmatrix}{{{\Pr\left\{ {{BX}_{1,1}X_{5,1}{E(1)}} \right\}} - {\Pr\left\{ {{BX}_{1,1}❘{E(1)}} \right\}}}} \\{+ {{{\Pr\left\{ {{BX}_{2,1}X_{5,1}{E(1)}} \right\}} - {\Pr\left\{ {{BX}_{2,1}❘{E(1)}} \right\}}}}} \\{+ {{{\Pr\left\{ {{BX}_{2,2}X_{5,1}{E(1)}} \right\}} - {\Pr\left\{ {{BX}_{2,2}❘{E(1)}} \right\}}}}}\end{pmatrix}}} \\{= 0.092519}\end{matrix}$ $\begin{matrix}{{\rho_{6}(1)} = {\frac{1}{m_{6}(1)}{\sum\limits_{k \in {{\{ 2\}}j}}{\sum\limits_{\in {S_{kJ}{(1)}}}{\sum\limits_{g \in {\{{1,2}\}}}{Pr\left\{ X_{ig} \middle| {E(1)} \right\}{{{\Pr\left\{ H_{kj} \middle| {X_{ig}{E(1)}} \right\}} - {Pr\left\{ H_{kj} \middle| {E(1)} \right\}}}}}}}}}} \\{= {\frac{1}{1}Pr\left\{ X_{6,1} \middle| {E(1)} \right\}\left( {{{{\Pr\left\{ {{BX}_{2,1}X_{6,1}{E(1)}} \right\}} - {\Pr\left\{ {{BX}_{2,1}❘{E(1)}} \right\}}}} + {{{\Pr\left\{ {{BX}_{2,2}X_{6,1}{E(1)}} \right\}} - {\Pr\left\{ {{BX}_{2,2}❘{E(1)}} \right\}}}}} \right)}} \\{{+ \frac{1}{1}}Pr\left\{ X_{6,2} \middle| {E(1)} \right\}\left( {{{{\Pr\left\{ {{BX}_{2,1}X_{6,2}{E(1)}} \right\}} - {\Pr\left\{ {{BX}_{2,1}❘{E(1)}} \right\}}}} + {{{\Pr\left\{ {{BX}_{2,2}X_{6,2}{E(1)}} \right\}} - {\Pr\left\{ {{BX}_{2,2}❘{E(1)}} \right\}}}}} \right)} \\{= 0.233261}\end{matrix}$ $\begin{matrix}{{\rho_{8}(1)} = {\frac{1}{m_{8}(1)}{\sum\limits_{k \in {{\{{1,2}\}}j}}{\sum\limits_{\in {S_{kJ}{(1)}}}{\sum\limits_{g \in {\{ 1\}}}{Pr\left\{ X_{ig} \middle| {E(1)} \right\}{{{\Pr\left\{ H_{kj} \middle| {X_{ig}{E(1)}} \right\}} - {Pr\left\{ H_{kj} \middle| {E(1)} \right\}}}}}}}}}} \\{= {\frac{1}{2}Pr\left\{ X_{8,1} \middle| {E(1)} \right\}\begin{pmatrix}{{{\Pr\left\{ {{BX}_{1,1}X_{8,1}{E(1)}} \right\}} - {\Pr\left\{ {{BX}_{1,1}❘{E(1)}} \right\}}}} \\{+ {{{\Pr\left\{ {{BX}_{2,1}X_{8,1}{E(1)}} \right\}} - {\Pr\left\{ {{BX}_{2,1}❘{E(1)}} \right\}}}}} \\{+ {{{\Pr\left\{ {{BX}_{2,2}X_{8,1}{E(1)}} \right\}} - {\Pr\left\{ {{BX}_{2,2}❘{E(1)}} \right\}}}}}\end{pmatrix}}} \\{= 0.065047}\end{matrix}$ $\begin{matrix}{{\rho_{9}(1)} = {\frac{1}{m_{9}(1)}{\sum\limits_{k \in {{\{{1,2}\}}j}}{\sum\limits_{\in {S_{kJ}{(1)}}}{\sum\limits_{g \in {\{ 1\}}}{Pr\left\{ X_{ig} \middle| {E(1)} \right\}{{{\Pr\left\{ H_{kj} \middle| {X_{ig}{E(1)}} \right\}} - {Pr\left\{ H_{kj} \middle| {E(1)} \right\}}}}}}}}}} \\{= {\frac{1}{2}Pr\left\{ X_{9,1} \middle| {E(1)} \right\}\begin{pmatrix}{{{\Pr\left\{ {{BX}_{1,1}X_{9,1}{E(1)}} \right\}} - {\Pr\left\{ {{BX}_{1,1}❘{E(1)}} \right\}}}} \\{+ {{{\Pr\left\{ {{BX}_{2,1}X_{9,1}{E(1)}} \right\}} - {\Pr\left\{ {{BX}_{2,1}❘{E(1)}} \right\}}}}} \\{+ {{{\Pr\left\{ {{BX}_{2,2}X_{9,1}{E(1)}} \right\}} - {\Pr\left\{ {{BX}_{2,2}❘{E(1)}} \right\}}}}}\end{pmatrix}}} \\{= 0.000199}\end{matrix}$ $\begin{matrix}{{\rho_{10}(1)} = {\frac{1}{m_{10}(1)}{\sum\limits_{k \in {{\{{1,2}\}}j}}{\sum\limits_{\in {S_{kJ}{(1)}}}{\sum\limits_{g \in {\{ 1\}}}{Pr\left\{ X_{ig} \middle| {E(1)} \right\}{{{\Pr\left\{ H_{kj} \middle| {X_{ig}{E(1)}} \right\}} - {Pr\left\{ H_{kj} \middle| {E(1)} \right\}}}}}}}}}} \\{= {\frac{1}{2}\frac{\Pr\left\{ X_{10,1} \middle| {E(1)} \right\}}{\Pr\left\{ {E(1)} \right\}}\begin{pmatrix}{{{\Pr\left\{ {{BX}_{1,1}X_{10,1}{E(1)}} \right\}} - {\Pr\left\{ {{BX}_{1,1}❘{E(1)}} \right\}}}} \\{+ {{{\Pr\left\{ {{BX}_{2,1}X_{10,1}{E(1)}} \right\}} - {\Pr\left\{ {{BX}_{2,1}❘{E(1)}} \right\}}}}} \\{+ {{{\Pr\left\{ {{BX}_{2,2}X_{10,1}{E(1)}} \right\}} - {\Pr\left\{ {{BX}_{2,2}❘{E(1)}} \right\}}}}}\end{pmatrix}}} \\{= 0.06229}\end{matrix}$The calculation results are listed in the following:

i ρ_(i)(1) 4 0 5 0.092519 6 0.233261 8 0.065047 9 0.000199 10 0.06299According to 6, the same as example 1, assume β_(i)=1/α_(i) and thevalues are follows:

i α_(i) β_(i) 4 100 0.01 5 2 0.5 6 50 0.02 8 2 0.5 9 2 0.5 10 2 0.5

Given m_(i)(1), λ_(i)(1) is calculated as follows:

i m_(i)(1) λ_(i)(1) 4 2 1/2 5 2 1/2 6 1 1/1 8 3 1/3 9 1 1 10 1 1

According to 8, take

${{I_{i}(y)} = \frac{{\lambda_{i}(y)}\beta_{i}{\rho_{i}(y)}}{\sum_{i \in S_{X{(y)}}}{{\lambda_{i}(y)}\beta_{i}{\rho_{i}(y)}}}},$we can obtain the following results:

${I_{4}(1)} = {\frac{{\lambda_{4}(1)}\beta_{4}{\rho_{4}(1)}}{\sum\limits_{i \in {\{{4,\cdots\mspace{14mu},9,10}\}}}{{\lambda_{i}(1)}\beta_{i}{\rho_{i}(1)}}} = {\frac{\frac{1}{2} \times {0.0}1 \times 0}{{0.0}70231} = 0}}$${I_{5}(1)} = {\frac{{\lambda_{5}(1)}\beta_{5}{\rho_{5}(1)}}{\sum\limits_{i \in {\{{4,\cdots\mspace{14mu},9,10}\}}}{{\lambda_{i}(1)}\beta_{i}{\rho_{i}(1)}}} = {\frac{\frac{1}{2} \times 0.5 \times 0.0231298}{0.070231} = {{0.3}29338}}}$${I_{6}(1)} = {\frac{{\lambda_{6}(1)}\beta_{6}{\rho_{6}(1)}}{\sum\limits_{i \in {\{{4,\cdots\mspace{14mu},9,10}\}}}{{\lambda_{i}(1)}\beta_{i}{\rho_{i}(1)}}} = {\frac{1 \times {0.0}2 \times 0.0046652}{{0.0}70231} = {{0.0}66427}}}$${I_{8}(1)} = {\frac{{\lambda_{8}(1)}\beta_{8}{\rho_{8}(1)}}{\sum\limits_{i \in {\{{4,\cdots\mspace{14mu},9,10}\}}}{{\lambda_{i}(1)}\beta_{i}{\rho_{i}(1)}}} = {\frac{\frac{1}{3} \times 0.5 \times {0.0}10841}{{0.0}70231} = {{0.1}54363}}}$${I_{9}(1)} = {\frac{{\lambda_{9}(1)}\beta_{9}{\rho_{9}(1)}}{\sum\limits_{i \in {\{{4,\cdots\mspace{14mu},9,10}\}}}{{\lambda_{i}(1)}\beta_{i}{\rho_{i}(1)}}} = {\frac{1 \times {0.5} \times 0.0000995}{{0.0}70231} = {{0.0}01417}}}$${I_{10}(1)} = {\frac{{\lambda_{10}(1)}\beta_{10}{\rho_{10}(1)}}{\sum\limits_{i \in {\{{4,\cdots\mspace{14mu},9,10}\}}}{{\lambda_{i}(1)}\beta_{i}{\rho_{i}(1)}}} = {\frac{1 \times 0.5 \times 0.031495}{{0.0}70231} = {{0.4}48449}}}$

The ranking results are:

index i I_(i)(1) 1 10 0.448449 2 5 0.329338 3 8 0.154363 4 6 0.066427 59 0.001417 6 4 0

According to 9, since X₅ is the only upstream variable of X₁₀, X₅ can beeliminated from the ranking. Since the rank probability of X₄ is equalto zero and there is no meaning to detect X₄, thus X₄ can be eliminatedfrom the ranking. So the above ranking becomes:

index i I_(i)(1) 1 10 0.448449 2 8 0.154363 3 6 0.066427 4 9 0.001417

Take the first three X-type variables into detection. Assume the resultsare X_(8,1), and X_(6,0) respectively. Then FIG. 9 becomes FIG. 12.Since α_(8,1;6,0)=a_(11,0;6,0)=“-”, which means no corresponding causalrelationship, FIG. 12 is simplified as FIG. 13 based on thesimplification rules, whereE(2)=E⁺(1)E(1)=X_(10,1)X_(8,1)X_(6,0)X_(7,1)X_(8,1)X_(10,1)X_(12,1). Tocalculate h_(1,1) ^(r)(2), h_(2,1) ^(r)(2), and h_(2,2) ^(r)(2), FIG. 13can be divided and simplified as FIG. 14 and FIG. 15. According to theDUCG algorithms in [8], we can obtain that:

Based on FIG. 14,

$\begin{matrix}{{\zeta_{1}(2)} = {{\Pr\left\{ {E(2)} \right\}} = {Pr\left\{ {X_{7,1}X_{8,1}X_{10,1}X_{12,1}} \right\}}}} \\{= {\Pr\left\{ {\left( {\frac{r_{7;4}}{r_{7}}A_{7,{1;4}}\frac{r_{4;1}}{r_{4}}A_{4;1}BX_{1}} \right)\begin{pmatrix}{\frac{r_{8;4}}{r_{8}}A_{8,{1;4}}\frac{r_{4;1}}{r_{4}}A_{4;1}BX_{1}} \\{{+ \frac{r_{8;5}}{r_{8}}}A_{8,{1;5}}\frac{r_{5;1}}{r_{5}}A_{5;1}BX_{1}}\end{pmatrix}\left( {\frac{r_{10;5}}{r_{10}}A_{10,{1;5}}\frac{r_{5;1}}{r_{5}}A_{5;1}BX_{1}} \right)X_{12,1}} \right\}}} \\{= {\Pr\begin{Bmatrix}{\left( {\frac{r_{7;4}}{r_{7}}A_{7,{1;4}}*\frac{r_{8;4}}{r_{8}}A_{8,{1;4}}\frac{r_{4;1}}{r_{4}}A_{4;1}} \right)*\left( {\frac{r_{10;5}}{r_{10}}A_{10,{1;5}}\frac{r_{5;1}}{r_{5}}A_{5;1}} \right){BX}_{1}X_{12,1}} \\{{+ \left( {\frac{r_{7;4}}{r_{7}}A_{7,{1;4}}\frac{r_{4;1}}{r_{4}}A_{4;1}} \right)}*\left( {\frac{r_{8;5}}{r_{8}}A_{8,{1;5}}*\frac{r_{10;5}}{r_{10}}A_{10,{1;5}}\frac{r_{5;1}}{r_{5}}A_{5;1}} \right){BX}_{1}X_{12,1}}\end{Bmatrix}}} \\{= {\Pr\left\{ {\begin{pmatrix}{\left( {\frac{r_{7;4}}{r_{7}}A_{7,{1;4}}*\frac{r_{8;4}}{r_{8}}A_{8,{1;4}}\frac{r_{4;1}}{r_{4}}A_{4;1}} \right)*\left( {\frac{r_{10;5}}{r_{10}}A_{10,{1;5}}\frac{r_{5;1}}{r_{5}}A_{5;1}} \right)} \\{{+ \left( {\frac{r_{7;4}}{r_{7}}A_{7,{1;4}}\frac{r_{4;1}}{r_{4}}A_{4;1}} \right)}*\left( {\frac{r_{8;5}}{r_{8}}A_{8,{1;5}}*\frac{r_{10;5}}{r_{10}}A_{10,{1;5}}} \right)}\end{pmatrix}{BX}_{1}X_{12,1}} \right\}}} \\{= {\Pr\left\{ {\begin{pmatrix}{\left( {\frac{1}{1}A_{7,{1;4}}\frac{1}{2}A_{8,{1;4}}\frac{1}{1}A_{4;1}} \right)*\left( {\frac{1}{1}A_{10,{1;5}}\frac{1}{1}A_{5;1}} \right)} \\{{+ \left( {\frac{1}{1}A_{7,{1;4}}\frac{1}{1}A_{4;1}} \right)}*\left( {\frac{1}{2}A_{8,{1;5}}*\frac{1}{1}A_{10,{1;5}}\frac{1}{1}A_{5;1}} \right)}\end{pmatrix}\left( {{\frac{r_{1;1}}{r_{1}}A_{1;1}B_{1}} + {\frac{r_{1;12}}{r_{1}}A_{{1;12},1}X_{12,1}}} \right)X_{12,1}} \right\}}} \\{= {\Pr\left\{ {\begin{pmatrix}{\left( {A_{7,{1;4}}*\frac{1}{2}A_{8,{1;4}}A_{4;1}} \right)*\left( {A_{10,{1;5}}A_{5;1}} \right)} \\{{+ \left( {A_{7,{1;4}}A_{4;1}} \right)}*\left( {\frac{1}{2}A_{8,{1;5}}*A_{10,{1;5}}A_{5;1}} \right)}\end{pmatrix}\left( {{\frac{1}{2}A_{1;1}B_{1}} + {\frac{1}{2}A_{{1;12},1}}} \right)X_{12,1}} \right\}}} \\{= {\Pr\left\{ {\begin{pmatrix}{\left( {A_{7,{1;4}}*\frac{1}{2}A_{8,{1;4}}A_{4;1}} \right)*\left( {A_{10,{1;5}}A_{5;1}} \right)} \\{{+ \left( {A_{7,{1;4}}A_{4;1}} \right)}*\left( {\frac{1}{2}A_{8,{1;5}}A_{5;1}*A_{10,{1;5}}} \right)}\end{pmatrix}\left( {{\frac{1}{2}A_{1;1}B_{1}} + {\frac{1}{2}A_{{1;12},1}}} \right)A_{12,{1;D}}} \right\}}} \\{= {\begin{pmatrix}{\left( {a_{7,{1;4}}*\frac{1}{2}a_{8,{1;4}}a_{4;1}} \right)*\left( {a_{10,{1;5}}a_{5;1}} \right)} \\{{+ \left( {a_{7,{1;4}}a_{4;1}} \right)}*\left( {\frac{1}{2}a_{8,{1;5}}*a_{10,{1;5}}a_{5;1}} \right)}\end{pmatrix}\left( {{\frac{1}{2}a_{1;1}b_{1}} + {\frac{1}{2}a_{{1;12},1}}} \right)a_{12,{1;D}}}} \\{= {\begin{pmatrix}{\left( {\begin{pmatrix} - & 0.7\end{pmatrix}*\frac{1}{2}\begin{pmatrix} - & 0.8\end{pmatrix}\begin{pmatrix} - & - \\ - & 0.9\end{pmatrix}} \right)*\left( \left( {\begin{pmatrix} - & 0.7\end{pmatrix}\begin{pmatrix} - & - \\ - & 0.7\end{pmatrix}} \right) \right)} \\{{+ \left( {\begin{pmatrix} - & 0.7\end{pmatrix}\begin{pmatrix} - & - \\ - & 0.9\end{pmatrix}} \right)}*\left( {\frac{1}{2}\begin{pmatrix} - & 0.7\end{pmatrix}*\begin{pmatrix} - & 0.7\end{pmatrix}\begin{pmatrix} - & - \\ - & 0.7\end{pmatrix}} \right)}\end{pmatrix}\left( {{\frac{1}{2}\begin{pmatrix} - & - \\ - & 1\end{pmatrix}\begin{pmatrix} - \\0.08\end{pmatrix}} + {\frac{1}{2}\begin{pmatrix} - \\0.7\end{pmatrix}}} \right)0.1}} \\{= {\left( {\left( {{- \ {0.1}}2348} \right) + \left( {{- \ {0.1}}08045} \right)} \right)\begin{pmatrix} - \\0.39\end{pmatrix}0.1}} \\{= {0.009029}}\end{matrix}$

Based on FIG. 15,

$\begin{matrix}{{\zeta_{2}(2)} = {{\Pr\left\{ {E(2)} \right\}} = {Pr\left\{ {X_{6,0}X_{7,1}X_{8,1}X_{10,1}X_{12,1}} \right\}}}} \\{= {\Pr\left\{ {\frac{r_{6;2}}{r_{6}}A_{6,{0;2}}BX_{2}\frac{r_{7,4}}{r_{7}}A_{7,{1;4}}\frac{r_{4;2}}{r_{4}}A_{4;2}B{X_{2}\begin{pmatrix}{\frac{r_{8;4}}{r_{8}}A_{8,{1;4}}\frac{r_{4;2}}{r_{4}}A_{4;2}BX_{2}} \\{{+ \frac{r_{8;5}}{r_{8}}}A_{8,{1;5}}\frac{r_{5;2}}{r_{5}}A_{5;2}{BX}_{2}}\end{pmatrix}}\frac{r_{10;5}}{r_{5}}A_{10,{1;5}}\frac{r_{5;2}}{r_{5}}A_{5;2}BX_{2}A_{12,{1;D}}} \right\}}} \\{= {\Pr\begin{Bmatrix}{\frac{r_{6;2}}{r_{6}}A_{6,{0;2}}*\left( {\frac{r_{7,4}}{r_{7}}A_{7,{1;4}}*\frac{r_{8;4}}{r_{8}}A_{8,{1;4}}\frac{r_{4;2}}{r_{4}}A_{4;2}} \right)*\left( {\frac{r_{10;5}}{r_{5}}A_{10,{1;5}}\frac{r_{5;2}}{r_{5}}A_{5;2}} \right){BX}_{2}A_{12,{1;D}}} \\{{+ \frac{r_{6;2}}{r_{6}}}A_{6,{0;2}}*\frac{r_{7,4}}{r_{7}}A_{7,{1;4}}\frac{r_{4;2}}{r_{4}}A_{4;2}*\left( {\frac{r_{8;5}}{r_{8}}A_{8,{1;5}}*A_{10,{1;5}}\frac{r_{5;2}}{r_{5}}A_{5;2}} \right){BX}_{2}A_{12,{1;D}}}\end{Bmatrix}}} \\{= {\Pr\left\{ {\begin{pmatrix}\left( {\frac{1}{1}A_{6,{0;2}}*\left( {\frac{1}{1}A_{7,{1;4}}*\frac{1}{2}A_{8,{1;4}}\frac{1}{1}A_{4;2}} \right)*\left( {\frac{1}{1}A_{10,{1;5}}\frac{1}{1}A_{5;2}} \right)} \right. \\{{+ \frac{1}{1}}A_{6,{0;2}}*\frac{1}{1}A_{7,{1;4}}\frac{1}{1}A_{4;2}*\left( {\frac{1}{2}A_{8,{1;5}}*\frac{1}{1}A_{10,{1;5}}\frac{1}{1}A_{5;2}} \right)}\end{pmatrix}{BX}_{2}A_{12,{1;D}}} \right\}}} \\{= {\Pr\left\{ {A_{6,{0;2}}*\begin{pmatrix}{\left( {A_{7,{1;4}}*\frac{1}{2}A_{8,{1;4}}A_{4;2}} \right)*\left( {A_{10,{1;5}}A_{5;2}} \right)} \\{{+ A_{7,{1;4}}}A_{4;2}*\left( {\frac{1}{2}A_{8,{1;5}}*A_{10,{1;5}}A_{5;2}} \right)}\end{pmatrix}A_{2;2}B_{2}A_{12,{1;D}}} \right\}}} \\{= {a_{6,{0;2}}*\begin{pmatrix}{\left( {a_{7,{1;4}}*\frac{1}{2}a_{8,{1;4}}a_{4;2}} \right)*\left( {a_{10,{1;5}}a_{5;2}} \right)} \\{{+ a_{7,{1;4}}}a_{4;2}*\left( {\frac{1}{2}a_{8,{1;5}}*a_{10,{1;5}}a_{5;2}} \right)}\end{pmatrix}a_{2;2}b_{2}a_{12,{1;D}}}} \\{= {\left( {{\begin{matrix}1 & {1 - 0.6 - 0.4}\end{matrix}1} - 0.3 - 0.7} \right)*\begin{pmatrix}\left( {\begin{pmatrix} - & 0.7\end{pmatrix}*\frac{1}{2}\begin{pmatrix} - & 0.8\end{pmatrix}\begin{pmatrix} - & - & - \\ - & 0.5 & 0.5\end{pmatrix}} \right) \\{*\left( {\begin{pmatrix} - & 0.7\end{pmatrix}\begin{pmatrix} - & - & - \\ - & 0.4 & 0.8\end{pmatrix}} \right)} \\{{+ \left( {\begin{pmatrix} - & 0.7\end{pmatrix}\begin{pmatrix} - & - & - \\ - & 0.5 & 0.5\end{pmatrix}} \right)}*} \\\left( {\frac{1}{2}\begin{pmatrix} - & 0.7\end{pmatrix}*\begin{pmatrix} - & 0.7\end{pmatrix}\begin{pmatrix} - & - & - \\ - & 0.4 & 0.8\end{pmatrix}} \right)\end{pmatrix}\begin{pmatrix} - & - & - \\ - & 1 & - \\ - & - & 1\end{pmatrix}\begin{pmatrix} - \\0.01 \\0.02\end{pmatrix}0.1}} \\{= 0}\end{matrix}$According to the algorithms in [8], the weighting coefficients

${\xi_{i}(y)} = \frac{\zeta_{i}(y)}{\sum\limits_{j}{\zeta_{j}(y)}}$of the sub-graphs are:

${{\xi_{1}(2)} = {\frac{\zeta_{1}(2)}{{\zeta_{1}(2)} + {\zeta_{2}(2)}} = {\frac{{0.0}09029}{{{0.0}09029} + 0} = 1}}}{{\xi_{2}(2)} = {\frac{\zeta_{2}(2)}{{\zeta_{1}(2)} + {\zeta_{2}(2)}} = {\frac{0}{{{0.0}09029} + 0} = 0}}}$Since ζ₂(0)=0 or ξ₂(0)=0, FIG. 15 does not hold and should beeliminated. FIG. 14 is the only valid sub-graph, which means, BX_(1,1)is the only event in S_(H)(2). According to 10, the ranking procedureends. The cause of abnormality of the current system is verified asBX_(1,1).

The invention claimed is:
 1. A method of efficiently diagnosing a realcause of a system abnormality by recommending an order of detectingstates of X-type variables, comprising: determining a ranking importancefactor corresponding to each of the X-type variables based at least inpart on determining a probability importance factor corresponding toeach of the X-type variables, wherein the X-type variables areassociated with possible causes of the system abnormality, and whereinthe probability importance factor corresponding to each of the X-typevariables is determined based on:${\rho_{i}(y)} = {\frac{1}{m_{i}(y)}{\sum\limits_{k \in {S_{iK}{(y)}}}{\omega_{k}{\sum\limits_{j \in {{S_{kJ}{(y)}}g} \in}{\sum\limits_{S_{iG}{(y)}}{Pr\left\{ X_{ig} \middle| {E(y)} \right\}{{{\Pr\left\{ H_{kj} \middle| {X_{ig}{E(y)}} \right\}} - {Pr\left\{ H_{kj} \middle| {E(y)} \right\}}}}}}}}}}$Or${\rho_{i}(y)} = {\frac{1}{m_{i}(y)}{\sum\limits_{k \in {S_{iK}{(y)}}}{\omega_{k}\sqrt{\sum\limits_{j \in {{S_{kJ}{(y)}}g} \in}{\sum\limits_{S_{iG}{(y)}}\left( {Pr\left\{ X_{ig} \middle| {E(y)} \right\}\left( {{Pr\left\{ H_{kj} \middle| {X_{ig}{E(y)}} \right\}} - {Pr\left\{ H_{kj} \middle| {E(y)} \right\}}} \right)} \right)^{2}}}}}}$Or${\rho_{i}(y)} = {\frac{1}{m_{i}(y)}{\sum\limits_{k \in {S_{iK}{(y)}}}{\omega_{k}{\sum\limits_{j \in {{S_{kJ}{(y)}}g} \in}{\sum\limits_{S_{iG}{(y)}}{{{\Pr\left\{ H_{kj} \middle| {X_{ig}{E(y)}} \right\}} - {Pr\left\{ H_{kj} \middle| {E(y)} \right\}}}}}}}}}$Or${\rho_{i}(y)} = {\frac{1}{m_{i}(y)}{\sum\limits_{k \in {S_{iK}{(y)}}}{\omega_{k}\sqrt{\sum\limits_{j \in {{S_{kJ}{(y)}}g} \in}{\sum\limits_{S_{iG}{(y)}}\left( {Pr\left\{ {{H_{kj}\left. {X_{ig}{E(y)}} \right\}} - {Pr\left\{ H_{kj} \right.{E(y)}}} \right\}} \right)^{2}}}}}}$Or${\rho_{i}(y)} = {\frac{1}{m_{i}(y)}{\sum\limits_{k \in {S_{iK}{(y)}}}{\omega_{k}{{{\sum\limits_{g \in {S_{iG}{(y)}}}{Pr\left\{ H_{kj} \middle| {X_{ig}{E(y)}} \right\}}} - {\frac{1}{m_{i}}{\sum\limits_{k \in {{S_{iK}{(y)}}j} \in {S_{kJ}{(y)}}}{\sum\limits_{g \in {S_{iG}{\langle y)}}}{Pr\left\{ H_{kj} \middle| {X_{ig}{E(y)}} \right\}}}}}}}}}}$wherein pi(y)represents a probability importance factor corresponding toa variable X_(i) indexed by i among the X-type variables in a y-th stepof state detection, mi(y) represents a number of states of the X_(i)variable in the y-th step, X_(ig) represents the variable X_(i) in astate g, S_(iG)(y) represents possible values of g in the y-th step,H_(kj) represents a possible cause of the system abnormality indexed byk in a state j, S_(ik)(y) represents possible values of k in the y-thstep, S_(kj) represents possible values of j in the y-th step, ω_(k)represents a danger degree of the H_(k) possible cause, and E(y)represents evidence comprising already known states of the X-typevariables in the y-th step; ranking the X-type variables based on theirrespective ranking importance factors; performing state detections forthe X-type variables based on a ranking list of the X-type variables;and determining the real cause of the system abnormality to takecorrecting actions so as to make the system normal based on results ofthe state detections associated with at least one subset of the X-typevariables.
 2. The method according to claim 1, further comprising:determining the ranking importance factor corresponding to each of theX-type variables based on the probability importance factor, a structureimportance factor, and a cost importance factor corresponding to each ofthe X-type variables; wherein the structure importance factorcorresponding to each of the X-type variables is determined based on thenumber of different possible causes with which each of the X-typevariables is associated; and wherein the cost importance factorcorresponding to each of the X-type variables is determined based on adifficulty degree, a waiting time, a price, and a damage degree ofperforming a state detection for each of the X-type variables.
 3. Themethod according to claim 2, wherein the ranking importance factorcorresponding to each of the X-type variables is determined based on:${I_{i}(y)} = \frac{{\lambda_{i}(y)}\beta_{i}{\rho_{i}(y)}}{\sum_{i \in {S_{X}{(y)}}}{{\lambda_{i}(y)}\beta_{i}{\rho_{i}(y)}}}$Or I_(i)(y) = λ_(i)(y)β_(i)ρ_(i)(y) Or${I_{i}(y)} = \frac{{\lambda_{i}(y)}{\rho_{i}(y)}}{\sum_{i \in {S_{X}{(y)}}}{{\lambda_{i}(y)}{\rho_{i}(y)}}}$Or I_(i)(y) = λ_(i)(y)ρ_(i)(y) Or${I_{i}(y)} = \frac{\beta_{i}{\rho_{i}(y)}}{\sum_{i \in {S_{X}{(y)}}}{\beta_{i}{\rho_{i}(y)}}}$Or I_(i)(y) = β_(i)ρ_(i)(y) Or${I_{i}(y)} = \frac{\rho_{i}(y)}{\sum_{i \in {S_{X}{(y)}}}{\rho_{i}(y)}}$Or I_(i)(y) = ρ_(i)(y) Or I_(i)(y) = w₁λ_(i)(y) + w₂ρ_(i)(y) + w₃β_(i)Or${I_{i}(y)} = \frac{{w_{1}{\lambda_{i}(y)}} + {w_{2}{\rho_{i}(y)}} + {w_{3}\beta_{i}}}{{\sum_{i \in {S_{X}{(y)}}}{w_{1}{\lambda_{i}(y)}}} + {w_{2}{\rho_{i}(y)}} + {w_{3}\beta_{i}}}$wherein I_(i)(y) represents a ranking importance factor corresponding tothe variable X_(i) among the X-type variables, λi(y) represents astructure importance factor corresponding to the variable X_(i) amongthe X-type variables, ρi(y) represents the probability importance factorcorresponding to the variable X_(i) among the X-type variables, β_(i)(y)represents a cost importance factor corresponding to the variable X_(i)among the X-type variables, and w₁, w₂, w₃ represents weight valuesassigned to the structure importance factor λi(y), the probabilityimportance factor ρi(y), and the cost importance factor β_(i)(y),respectively.
 4. The method according to claim 1, further comprising:determining a structure importance factor corresponding to each of theX-type variables based on a number of different possible causes withwhich each of the X-type variables is associated, and determining theranking importance factor corresponding to each of the X-type variablesbased at least in part on the probability importance factor and thestructure importance factor corresponding to each of the X-typevariables.
 5. The method according to claim 1, further comprising:assigning a value to each of a plurality of states corresponding to eachof the possible causes based on their respective danger degrees.
 6. Themethod according to claim 1, further comprising: determining a costimportance factor corresponding to each of the X-type variables based ona difficulty degree, a waiting time, a price, and a damage degree ofperforming a state detection for each of the X-type variables.
 7. Themethod according to claim 1, further comprising: in response todetermining that a first variable with a higher rank is an ancestor ordescendant variable of a second variable with a lower rank, eliminatingthe second variable from the ranking list of the X-type variables,wherein the first variable and the second variable are among the X-typevariables; and in response to determining that any particular variableamong the X-type variables has a ranking importance factor equal tozero, eliminating the any particular variable from the ranking list ofthe X-type variables.